Flag Hardy space theory on Heisenberg groups and applications
Peng Chen, Michael G. Cowling, Ming-Yi Lee, Ji Li, Alessandro Ottazzi
aa r X i v : . [ m a t h . F A ] F e b Flag Hardy space theory on Heisenberg groups and applications
Peng Chen, Michael G. Cowling, Ming-Yi Lee, Ji Li and Alessandro Ottazzi
Abstract
We establish a complete theory of the flag Hardy space H F ( H ν ) on the Heisenberg group H ν with characterisations via atomic decompositions, area functions, square functions, maximalfunctions and singular integrals. We introduce several new techniques to overcome the difficultiescaused by the noncommutative Heisenberg group multiplication, and the lack of a suitable Fouriertransformation and Cauchy–Riemann type equations. Applications include the boundedness from H F ( H ν ) to L ( H ν ) of various singular integral operators that arise in complex analysis, a sharpboundedness result on H F ( H ν ) of the Marcinkiewicz-type multipliers introduced by M¨uller, Ricciand Stein, and the decomposition of flag BMO space via singular integrals. Contents H ν and cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5 Estimates for the heat and Poisson kernels . . . . . . . . . . . . . . . . . . . . . . . . . 232.6 Flag Riesz transforms and singular integrals . . . . . . . . . . . . . . . . . . . . . . . . 252.7 Flag Sobolev inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.8 Equivalence of various square functions . . . . . . . . . . . . . . . . . . . . . . . . . . 26 H F, atom ( H ν ) ⊆ H F, area ( H ν ) . . . . . . . . . . . . . . . . . . . . . . . . . 373.5 The inclusion H F, atom ( H ν ) ⊆ H F, cts ( H ν ) . . . . . . . . . . . . . . . . . . . . . . . . . . 453.6 The inclusion H F, atom ( H ν ) ⊆ H F, dis ( H ν ) . . . . . . . . . . . . . . . . . . . . . . . . . . 461.7 The inclusion H F, atom ( H ν ) ⊆ H F, gmax ( H ν ) . . . . . . . . . . . . . . . . . . . . . . . . . 463.8 Simple singular integral operators on the atomic Hardy space . . . . . . . . . . . . . . 52 H F, area ( H ν ) ⊆ H F, atom ( H ν ) . . . . . . . . . . . . . . . . . . . . . . . . . 574.3 Remarks on a “discrete area function” . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 H F, cts ( H ν ) ⊆ H F, area ( H ν ) . . . . . . . . . . . . . . . . . . . . . . . . . . 615.2 The inclusion H F, dis ( H ν ) ⊆ H F, atom ( H ν ) . . . . . . . . . . . . . . . . . . . . . . . . . . 62 H F, nontan ( H ν ) ⊆ H F, area ( H ν ) . . . . . . . . . . . . . . . . . . . . . . . . . 646.2 The inclusion H F, radial ( H ν ) ⊆ H F, nontan ( H ν ) . . . . . . . . . . . . . . . . . . . . . . . . 70 H F, Riesz ( H ν ) ⊆ H F, dis ( H ν ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727.2 Conclusion of the proof of Theorem 1.1 and remarks . . . . . . . . . . . . . . . . . . . 75 H F ( H ν ) via heat semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758.2 Comparison with the Han–Lu–Sawyer Hardy space H HLS ( H ν ) . . . . . . . . . . . . . 768.3 H F ( H ν ) is a proper subspace of the Hardy space H F SCG ( H ν ) . . . . . . . . . . . . . . 768.4 Sharp endpoint boundedness of certain flag singular integrals . . . . . . . . . . . . . . 778.5 Sharp endpoint boundedness of Marcinkiewicz multipliers m (∆ (1) , i T ) . . . . . . . . . 78 In 1999, Washington University in Saint Louis hosted a conference on Harmonic Analysis to celebratethe 70th birthday of G. Weiss. In his talk in flag singular integral operators, E. M. Stein asked “Whatis the Hardy space theory in the flag setting?”In this paper, we characterise completely a flag Hardy space H F ( H ν ) on the Heisenberg group H ν .It is a proper subspace of the classical one-parameter Hardy space of Folland and Stein [23] that wasstudied by Christ and Geller [8]. Our space is useful in several applications:
1) endpoint boundedness of certain singular integrals , including the Hilbert transform in thecentral variable, the homogeneous kernels considered first by Folland and Stein [18], the Cauchy–Szeg¨oprojection on H ν and the singular integrals considered by Phong and Stein [54] in their study of the¯ ∂ -Neumann problem (see also [58]);
2) endpoint boundedness of two-parameter Marcinkiewicz multipliers as studied by Muller,Ricci and Stein [46, 47], and the one-parameter multipliers associated to the sub-Laplacian in theHeisenberg group considered by Hebisch [33] and by M¨uller and Stein [48];
3) representation of functions in the flag BMO space on H ν .2his opens the door to finding complete characterisations of Hardy spaces on more general ho-mogeneous Lie groups with implicit multi-parameter structures, and has potential applications to thestudy of the ¯ ∂ b -complex on different domains, continuing the path blazed by Nagel, Ricci and Stein[50] and Nagel, Ricci, Stein and Wainger [51, 52], and more generally to the development of the L p theory of flag singular integral operators on H ν and other nilpotent Lie groups.In this section, we present the background to our results and the difficulties, state our results,outline the rest of the paper and explain what is new in our work. Harmonic analysis has had a profound influence in partial differential equations and several complexvariables. Current approaches to one-parameter harmonic analysis have been developed from the 1950son; the Calder´on–Zygmund theory of singular integrals and the related function spaces are central tothis theory. In the setting of Euclidean spaces R n , a notable property of standard Calder´on–Zygmundsingular integral operators and also the Hardy–Littlewood maximal operator, is that they commutewith the classical dilations δ t , given by δ t x = ( tx , . . . , tx n ) for all x ∈ R n and all t > R n ), hasbeen studied extensively since the 1970s by Gundy, Stein, Chang, R. Fefferman, Journ´e, Pipher,Lacey, and many others (see, for example, [4, 5, 6, 19, 20, 21, 26, 39, 53]). To show the boundedness ofCalder´on–Zygmund singular integrals, there are corresponding function spaces, notably Hardy spacesand their duals, which provide a natural way to do this easily. These space were developed in [9, 18]in the one-parameter setting, and later in [4, 39] in the multi-parameter product setting.A more recent breakthrough is due to M¨uller, Ricci and Stein [46]. They introduced a new typeof multi-parameter structure, which lies between the one-parameter and multi-parameter cases, andstudied the L p boundedness of Marcinkiewicz multiplier operators m (∆ (1) / |T | , i T ) on the Heisenberggroup H ν , where ∆ (1) is the sub-Laplacian on H ν , T is the central invariant vector field, and m is a multiplier of Marcinkiewicz type. They proved the L p boundedness of m (∆ (1) / |T | , i T ) by usinglifting and projection arguments when m satisfies C ∞ regularity conditions. Using the same approach,they also establish the L p boundedness of certain singular integrals studied by Phong and Stein [54]that arise in the ¯ ∂ -Neumann problem (see also [58]). The new multi-parameter structure, called a flag structure , is implicit, that is, it cannot be written in terms of explicit dilations, and this leadsto completely new difficulties that do not appear in the one-parameter or product settings. Theimplicit structure is obtained by taking the product space H ν × R , and identifying points (( z, t ) , s ) and(( z ′ , t ′ ) , s ′ ) when z = z ′ and t + s = t ′ + s ′ . Under this identification, a product of balls B × B in theproduct space H ν × R becomes a group theoretic product B · B of sets in H ν . The identification givesrise to a projection from functions on H ν × R to functions on H ν : let F = F (cid:0) ( z, t ) , s (cid:1) be a functionon H ν × R , with ( z, t ) ∈ H ν and s ∈ R , and define the projected function f by f ( z, t ) = Z R F (cid:0) ( z, t − s ) , s (cid:1) ds. See also the survey paper by Stein [58]. More recently, Nagel, Ricci and Stein [50] studied a class ofoperators on nilpotent Lie groups G given by convolution with flag kernels and applied this theory to3tudy the (cid:3) b -complex on certain CR submanifolds of C n . Recently, Nagel, Ricci, Stein and Wainger[51, 52] developed the theory of singular integrals with flag kernels in the more general setting ofhomogeneous groups. They proved that singular integral operators on these groups with flag kernelsare bounded on L p when 1 < p < ∞ , and form an algebra. See also the recent results of Street [60].Motivated by Stein’s original question in 1999, the following questions arise naturally along thisline [46, 58, 50, 51, 52] of the study of multi-parameter flag structure arising from ¯ ∂ b Neumann problemin several complex variables and from multi-parameter structure on group theory.
Question 1.
Is there a flag Hardy space H F ( H ν ) on the Heisenberg group that may be characterisedin terms of square functions, maximal functions, atomic decompositions, and Riesz transforms? Question 2.
What is the relationship of the space H F ( H ν ) to the one-parameter Hardy space H F SCG ( H ν ) introduced by Folland and Stein [23] and then studied by Christ and Geller [8]? Question 3.
There are various singular integrals on the Heisenberg group that appear in connectionwith boundary value problems in complex analysis in several variables and that are known to bebounded on the spaces L p ( H ν ) when 1 < p < ∞ . These include the Cauchy–Szeg˝o projection, whichhas a homogeneous kernel, and the operators introduced in [54], which have nonhomogeneous kernels.Are these operators also bounded from H F ( H ν ) to L ( H ν )? Question 4.
Suppose that the Marcinkiewicz multiplier function m satisfies the sharp regularityconditions α > ν and β > / m (∆ (1) , i T ) is bounded on L p ( H ν ) when 1 < p < ∞ . Is m (∆ (1) , i T ) is bounded on H F ( H ν )? Question 5.
Does the dual space of H F ( H ν ) have a decomposition like that of the one-parameterspace BMO( H ν ) of [23]? Answers.
We answer Question 1 by describing a Hardy space H F ( H ν ) which may be characterised byatomic decompositions, square functions, area functions, maximal functions and flag Riesz transforms.From the maximal function characterisations of H F ( H ν ) and of H F SCG ( H ν ), it is apparent that H F ( H ν ) ⊆ H F SCG ( H ν ). We also show that H F ( H ν ) is a proper subspace of H F SCG ( H ν ), whichaddresses Question 2.Next, we give a positive answer to Question 3, by verifying that kK a k L ( H ν ) is uniformly boundedwhen a is a flag atom. A similar approach confirms the boundedness of the Cauchy–Szeg¨o operatoron H ν , even though the corresponding kernel does not satisfy the cancellation condition.Moreover, we prove that the Marcinkiewicz multiplier operator m (∆ (1) , i T ) is bounded on H F ( H ν )when m satisfies the sharp regularity condition above, thereby answering Question 4. In particular,we see that when m satisfies the sharp regularity condition, the one-parameter H¨ormander multiplieroperator m (∆ (1) ) is bounded on H F ( H ν ), and hence also on H F SCG ( H ν ) which was apparently notknown before. By interpolation we also find another proof of the result of M¨uller–Stein [48] andHebisch [33].Finally, we note that the characterisation of H F ( H ν ) via flag Riesz transforms and the duality of H F ( H ν ) with flag BMO imply that BMO F ( H ν ) has a decomposition. Remarks.
We point out that Han, Lu and Sawyer [32] defined a flag Hardy space H HLS ( H ν ) using adiscrete Littlewood–Paley square function, with a convolution of the form ψ m,n ∗ f , and described theinterpolation spaces between H HLS ( H ν ) and L ( H ν ). We use their interpolation theorem by finding4n isomorphism between our space and theirs. They dealt with H p spaces when p <
1, so their workis more general than ours; however, the heat and Poisson semigroups and Riesz transforms are notgiven by left convolutions, so their methods do not link to these standard operators.Han, Lu and Sawyer also proved the boundedness of singular integral operators from H HLS ( H ν ) to L ( H ν ) provided that the kernel satisfies a cancellation condition. However, the kernel of the Cauchy–Szeg¨o projection does not satisfy the cancellation condition, and their methods cannot handle the radialand nontangential maximal functions or the operators m (∆ (1) ) of M¨uller–Stein [48] and Hebisch [33]and m (∆ (1) , i T ) of M¨uller–Ricci–Stein [46], which do not even have an explicit kernel. Our atomicdecomposition shows that all these operators are bounded from H F ( H ν ) to L ( H ν ).Recently, Han, Wick, and the third and fourth authors [28] gave a full description of the flag Hardyspace on the simplified model R m × R n ; their main tools include Fourier transforms, Cauchy–Riemannequations and the geometrical fact that a flag rectangle R on R m × R n can be written as a product R = I × J , where I and J are cubes in R m and R n . These tools and geometry do not apply in theflag setting on H ν .Further progress on related problems in the flag setting includes (but is not limited to) [13, 27, 30,31, 59, 64, 65]. In this section, we state our main results in more detail.We begin with a little notation; the details and further notation are given in the next section. TheHeisenberg group H ν is a Lie group, with underlying manifold C ν × R ; a typical element is written g ,or in coordinates, as ( z, t ), where z ∈ C ν and t ∈ R . We may also write ( z, t ) as ( z , . . . , z ν , t ), or as( x, y, t ), or as ( x , . . . , x ν , y , . . . , y ν , t ). The multiplication law is( z ′ , t ′ ) · ( z, t ) = ( z ′ + z, t ′ + t + S ( z ′ , z )) , where S is the (slightly nonstandard) symplectic form on C ν given by S ( z ′ , z ) = 4 ν Im (cid:18) ν X j =1 z ′ j ¯ z j (cid:19) = 4 ν (cid:0) h y ′ , x i − h x ′ , y i (cid:1) = 4 ν (cid:18) ν X j =1 y ′ j x j − ν X j =1 x ′ j y j (cid:19) . The identity of H ν is written o or (0 , z, t ) − = ( − z, − t ). The Haar measure on H ν is the Lebesgue measure, which we write dg or dz dt . The standard Heisenberg group convolutionis given by the formula f ∗ (1) f ( g ) = Z H ν f ( g ) f ( g − g ) dg = Z H ν f ( gg ) f ( g − ) dg ∀ g ∈ H ν . The flag structure on H ν involves the subgroup { (0 , t ) : t ∈ R } , which we may identify with R inthe obvious way (unfortunately this may be a little confusing). We convolve a function f on H ν witha function f on R as follows: f ∗ (2) f ( g ) = Z R f ( g ) f ( g − g ) dg ∀ g ∈ H ν . (Thus we think of functions on R as distributions on H ν which are supported in the centre of H ν .)5he ball with centre g and radius r in the gauge metric on H ν (see Section 2.2 for the definition)is denoted by B (1) ( g , r ); it coincides with g · B (1) ( o, r ). The interval ( g − s, g + s ) is denoted by B (2) ( g , s ) (and identified with a subset of H ν ). As mentioned above, the basic geometric object in theflag structure, analogous to the ball in classical analysis and to the direct product of balls in analysison product spaces, is the (group) product of balls B (1) ( g , r ) · B (2) ( g , s ).We write ∆ (1) for the usual sub-Laplacian on H ν and ∆ (2) for the Laplacian on R , both normalisedto be positive operators, and T for the central invariant vector field on R . These operators, which areinterpreted distributionally, and the associated geometry are key to our results.Now, we give seven different alternative definitions of flag Hardy spaces. First, given an opensubset Ω of H ν , we write M (Ω) for the set of all maximal adapted subrectangles of Ω (see the nextsection for detailed notation and definitions). Definition.
Fix integers
M > ν/ N ≥ κ >
1. An atom is a function a ∈ L ( H ν ) such that there exists an open subset Ω of H ν of finite measure | Ω | and functions a R in L ( H ν ), called particles , and b R in Dom(∆ (1) M ∆ (2) N ) in L ( H ν ) for all R ∈ M (Ω) such that(A1) a R = ∆ (1) M ∆ (2) N b R and supp b R ⊆ R ∗ , where R ∗ is a κ -enlargement of R ;(A2) for all sign sequences σ : M (Ω) → {± } , the sum P R ∈ M (Ω) σ R a R converges in L ( H ν ), to a σ say, and k a σ k L ( H ν ) ≤ | Ω | − / ;(A3) a = P R ∈ M (Ω) a R .We say that f ∈ L ( H ν ) has an atomic decomposition if we may write f as a sum P j ∈ N λ j a j ,converging in L ( H ν ), where P j ∈ N | λ j | < ∞ and each a j is an atom; we write f ∼ P j ∈ N λ j a j toindicate that P j ∈ N λ j a j is an atomic decomposition of f . The space H F, atom ( H ν ) is defined to be thelinear space of all f ∈ L ( H ν ) that have atomic decompositions, with norm k f k H F, atom ( H ν ) := inf (cid:26) X j ∈ N | λ j | : f ∼ X j ∈ N λ j a j (cid:27) . We provide a more complete definition at the beginning of Section 3, and in particular make thenotion of a κ -enlargement R ∗ of a rectangle R precise.Several characterisations of classical Hardy spaces involve integrals or maximal functions overcones. Given g ∈ H ν and β >
0, we define the cone Γ β ( g ) as follows:Γ β ( g ) := { ( g ′ , r, s ) ∈ H ν × R + × R + : g ′ ∈ g · B (1) ( o, βr ) · B (2) (0 , β s ) } . (1.1)For simplicity, we write Γ ( g ) as Γ( g ). It may be shown that changing the parameter β does notchange the Hardy space, though it changes the norm to an equivalent norm.For our next definition, we take a function ˜ ϕ (1) on H ν and a function ˜ ϕ (2) on R , satisfying varioussmoothness, decay and spectral conditions, and we consider ϕ (1) = ∆ M (1) ˜ ϕ (1) and ϕ (2) = ∆ N (2) ˜ ϕ (2) , where M > ν/ N ≥
1. Further, we define χ (1) r to be the L ( H ν )-normalised characteristic function ofthe ball B (1) ( o, βr ) and χ (2) s to be the L ( R )-normalised characteristic function of the ball B (2) (0 , β s ). Definition.
Suppose that ϕ (1) , ϕ (2) , χ (1) and χ (2) are as above. For f ∈ L ( H ν ), we define theLusin–Littlewood–Paley area function S F, area , ϕ ( f ) associated to ϕ (1) and ϕ (2) by S F, area , ϕ ( f )( g ) := (cid:18)Z Z R + × R + (cid:12)(cid:12) f ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s (cid:12)(cid:12) ∗ (1) χ (1) r ∗ (2) χ (2) s ( g ) drr dss (cid:19) / g ∈ H ν , and we define the Hardy space H F, area , ϕ ( H ν ), usually abbreviated to H F, area ( H ν ), tobe the set of all f ∈ L ( H ν ) for which S F, area , ϕ ( f ) ∈ L ( H ν ), with norm k f k H F, area , ϕ ( H ν ) := kS F, area , ϕ ( f ) k L ( H ν ) . In Section 4, we shall make the conditions on ˜ ϕ (1) and ˜ ϕ (2) precise, and we shall see that theLusin–Littlewood–Paley Hardy space is independent of the choice of ˜ ϕ (1) and ˜ ϕ (2) . In particular, wemay take ˜ ϕ (1) and ˜ ϕ (2) to be the heat kernels or the Poisson kernels associated to ∆ (1) and ∆ (2) respectively. Definition.
Suppose that ϕ (1) and ϕ (2) are as above. For f ∈ L ( H ν ), we define the continuous anddiscrete Littlewood–Paley square functions S F, cts , ϕ ( f ) and S F, dis , ϕ ( f ) associated to ϕ (1) and ϕ (2) by S F, cts , ϕ ( f )( g ) := (cid:18)Z Z R + × R + (cid:12)(cid:12) f ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s ( g ) (cid:12)(cid:12) drr dss (cid:19) / S F, dis , ϕ ( f )( g ) := (cid:18) X ( m,n ) ∈ Z × Z (cid:12)(cid:12) f ∗ (1) ϕ (1)2 m ∗ (2) ϕ (2)2 n ( g ) (cid:12)(cid:12) (cid:19) / for all g ∈ H ν . We define the square function Hardy spaces H F, cts , ϕ ( H ν ) and H F, dis , ϕ ( H ν ), often abbre-viated to H F, cts ( H ν ) and H F, dis ( H ν ), to be the sets of all f ∈ L ( H ν ) for which S F, cts , ϕ ( f ) ∈ L ( H ν )or S F, dis , ϕ ( f ) ∈ L ( H ν ), with norms k f k H F, cts , ϕ ( H ν ) := kS F, cts , ϕ ( f ) k L ( H ν ) , k f k H F, dis , ϕ ( H ν ) := kS F, dis , ϕ ( f ) k L ( H ν ) . In Section 5, we shall make the conditions on ˜ ϕ (1) and ˜ ϕ (2) precise, and we shall see that the squarefunction Hardy space is largely independent of the choice of ˜ ϕ (1) and ˜ ϕ (2) . In particular, we may take˜ ϕ (1) and ˜ ϕ (2) to be the heat kernels or the Poisson kernels associated to ∆ (1) and ∆ (2) respectively.Up to a modification, to be discussed later, our space H F, dis ( H ν ) agrees with the Hardy space of Han,Lu and Sawyer [32].Next, let p (1) r be the Poisson kernel, that is, the convolution kernel of the operator e − r √ ∆ (1) on H ν ,and p (2) s be the standard Poisson kernel e − s √ ∆ (2) on R (here r, s ∈ R + ). We define the flag Poissonkernel as follows: p r,s ( g ) = p (1) r ∗ (2) p (2) s ( g ) = Z R p (1) r ( z, t − t ′ ) p (2) s ( t ′ ) dt ′ , (1.2)where g = ( z, t ) ∈ H ν , and the flag Poisson integral u (or u ( f )) of f ∈ L ( H ν ) by u ( g, r, s ) := f ∗ (1) p r,s ( g ) . Definition.
Fix β >
0. The nontangential maximal function of f ∈ L ( H ν ) is defined by u ∗ ( g ) = u ∗ ( f )( g ) = sup ( g ′ ,r,s ) ∈ Γ β ( g ) | u ( g ′ , r, s ) | . The space H F, nontan ( H ν ) is defined to be the linear space of all f ∈ L ( H ν ) such that u ∗ ∈ L ( H ν ),with norm k f k H F, nontan ( H ν ) := k u ∗ ( f ) k L ( H ν ) . Definition.
The radial maximal function of f ∈ L ( H ν ) is defined by u + ( g ) = u + ( f )( g ) = sup r,s ∈ R + | u ( g ′ , r, s ) | . The space H F, radial ( H ν ) is defined to be the set of all f ∈ L ( H ν ) such that u + ∈ L ( H ν ), with norm k f k H F, radial ( H ν ) := k u + ( f ) k L ( H ν ) . F ( H ν ) of pairs ϕ of functions ϕ (1) on H ν and ϕ (2) on R that,together with their derivatives, behave like Poisson kernels. The precise conditions are stated later,in Definition 2.20. Definition.
The grand maximal function M F, gmax ( f ) of f ∈ L ( H ν ) is defined by M F, gmax ( f )( g ) = sup ϕ ∈ F ( H ν ) sup r,s ∈ R + (cid:12)(cid:12) f ∗ (1) ϕ r,s ( g ) (cid:12)(cid:12) ∀ g ∈ H ν . The space H F, gmax ( H ν ) is defined to be the set of all f ∈ L ( H ν ) such that M F, gmax ( f ) is in L ( H ν ),with norm k f k H F, gmax ( H ν ) := kM F, gmax ( f ) k L ( H ν ) . Our final definition brings in singular integral operators.
Definition.
The (tensor-valued) flag Riesz transformation is defined by R F = ∇ (1) ∆ − / ∇ (2) ∆ − / , and the space H F, Riesz ( H ν ) is the set of all f ∈ L ( H ν ) such that ∇ (1) ∆ − / ( f ), ∇ (2) ∆ − / ( f ) and R F ( f ) all lie in L ( H ν ), with norm k f k H F, Riesz ( H ν ) := k f k L ( H ν ) + (cid:13)(cid:13)(cid:13) ∇ (1) ∆ − / ( f ) (cid:13)(cid:13)(cid:13) L ( H ν ) + (cid:13)(cid:13)(cid:13) ∇ (2) ∆ − / ( f ) (cid:13)(cid:13)(cid:13) L ( H ν ) + kR F ( f ) k L ( H ν ) . The main result of this paper is that the definitions above all agree.
Theorem 1.1.
The spaces defined above all coincide, that is, H F, atom ( H ν ) = H F, area ( H ν ) = H F, cts ( H ν ) = H F, dis ( H ν ) = H F, nontan ( H ν )= H F, radial ( H ν ) = H F, gmax ( H ν ) = H F, Riesz ( H ν ) . Moreover, they have equivalent norms, and the constants in the equivalences depend only on M , N , κ , β and ν . In light of this theorem, we now use the notation H F ( H ν ) to denote any of these spaces. (1) To connect our results with previous work, we relate our space H F ( H ν ) with the Hardy space H HLS ( H ν ) of Han, Lu and Sawyer [32]. Proposition 1.2.
The spaces H F ( H ν ) and H HLS ( H ν ) are isomorphic and have equivalent norms. This proposition will be proved in Section 8.2. It then follows from [32] that L p ( H ν ) is the complexinterpolation space between H F ( H ν ) and L ( H ν ), and BMO F ( H ν ) is the dual space of H F ( H ν ). (2) Folland and Stein [23] defined a one-parameter Hardy space on stratified Lie groups, and char-acterised it by Littlewood–Paley theory, atomic and molecular decompositions, and tangential andnontangential maximal functions. Later Christ and Geller [8] characterised this Hardy space by sin-gular integrals, and in particular by Riesz transforms. We denote this one-parameter Hardy spaceby H F SCG ( H ν ). From our Theorem 1.1, it follows that our space H F ( H ν ) is a proper subspace of H F SCG ( H ν ). 8 roposition 1.3. H F ( H ν ) ( H F SCG ( H ν ) . The proof of this proposition will be given in Section 8.3. (3)
Consider the Phong-Stein singular integral operator K , which arises in solving the ¯ ∂ -Neumannproblem in a bounded smooth domain in C ν +1 (see Phong and Stein [54], [46, Section 5] or [58]). Itis a convolution operator, that is, K f = f ∗ k P S , and the convolution kernel k P S is given by k P S ( z, t ) = ω ( z )( | z | + t ) ν | z | + it ∀ ( z, t ) ∈ H ν , (1.3)where ω is homogeneous of degree 0 on C ν , smooth away from the origin and with mean-value 0 onthe unit sphere. Stein [58] proved that K is bounded on L p ( H ν ) when 1 < p < ∞ by a lifting andprojection argument.Another important singular integral operator on H ν is the Cauchy–Szeg¨o projection C , which givesan analytic function in the Siegel upper half space in terms of its boundary value. Its restriction tothe boundary is a convolution operator, that is, C ( f ) = f ∗ k CS , and the convolution kernel k CS isgiven by k CS ( z, t ) = c ( | z | + it ) ν +1 ∀ ( z, t ) ∈ H ν (1.4)(see [57, Chapter12, Section 2.4]).By using the atomic decomposition of H F ( H ν ), we will obtain endpoint estimates for both theseoperators in Section 8.4. Proposition 1.4.
The Phong-Stein operator K is bounded from H F ( H ν ) to L ( H ν ) . Proposition 1.5.
The Cauchy–Szeg¨o operator C is bounded from H F ( H ν ) to L ( H ν ) . We point out that the cancellation of ω is only used to show the L boundedness of K (whichfollows from the Cotlar–Stein almost orthogonality estimate [54, Theorem 4]). When using the atomicdecomposition, H F ( H ν )– L ( H ν ) boundedness follows from L boundedness and decay estimates onthe kernel, but does not require any cancellation condition on the kernel. The boundedness of theCauchy–Szeg¨o operator reflects this; its kernel has no cancellation condition. (4) Define the two-parameter Sobolev space L α,β ( R × R + ) to be the collection of tempered distributions m on R for which the following norm is finite: k m k L α,β ( R ) := (cid:18)Z Z R (1 + | ξ | ) α (1 + | ξ | + | ξ | ) β | ˆ m ( ξ , ξ ) | dξ dξ (cid:19) / ; (1.5)here ˆ m denotes the usual Fourier transform of m .Let m r,s be the two-parameter dilate of m , that is, m r,s ( ξ , ξ ) := m ( rξ , sξ ). Choose η ∈ C ∞ c ( R + )with support in (1 / ,
2) such that P j ∈ Z η ( · / j ) = 1, and define η , ( ξ , ξ ) := η ( ξ ) η ( | ξ | ). Then wehave the following sharp result on the endpoint boundedness of Marcinkiewicz multipliers on H ν . Proposition 1.6.
If the function m satisfies the condition sup r,s> k η , m r,s k L α,β ( R ) < ∞ , where α > ν and β > / , then the joint spectral multiplier m (∆ (1) / |T | , i T ) is bounded on H F ( H ν ) . The L p boundedness of m (∆ (1) , i T ) was proved when 1 < p < ∞ in [47]. The proof of thisproposition will be given in Section 8.5. 9ven for the classical one-parameter Mihlin–H¨ormander multiplier, when we consider the sharpindex of differentiation, there is no pointwise estimate for the kernel, see [57]).Fix a nontrivial, smooth, nonnegative-real-valued function η on R + with support in [1 / , α > (2 ν + 1) /
2. Write L α ( R ) for the usual Sobolev space on R , whose norm is defined by k m k L α ( R ) := (cid:18)Z Z R (1 + | ξ | ) α ˆ m ( ξ ) | dξ (cid:19) / . M¨uller and Stein [48] and Hebisch [33] showed that the H¨ormander multiplier m (∆ (1) ) is bounded on L p ( H ν ) for 1 < p < ∞ , if sup t> k η ( · ) m ( t · ) k L α ( R ) < ∞ .An interesting fact that follows from Proposition 1.6 is that, this one parameter H¨ormander mul-tiplier m (∆ (1) ) is also bounded on our multi-parameter flag Hardy space. To be more precise, Corollary 1.7.
If the function m satisfies the condition sup t> k η ( · ) m ( t · ) k L α ( R ) < ∞ , where α > (2 ν + 1) / , then m (∆ (1) ) is bounded on H F ( H ν ) . By interpolation and duality, m (∆ (1) ) is bounded on L p ( H ν ) for 1 < p < ∞ . This provides anotherproof of the result of M¨uller and Stein [48] and of Hebisch [33]. (5) From [32], it is known that the duality of H HLS ( H ν ) is BMO F ( H ν ) (see Definition 6 and Theorem7 in [32] for full details). From Theorem 1.1 and Proposition 1.2, we obtain a decomposition ofBMO F ( H ν ). Proposition 1.8.
The following are equivalent:1. b ∈ BMO F ( H ν ) ;2. there exist (vector-valued) functions h , h , h and h in L ∞ ( H ν ) such that b = h + ∇ (1) ∆ − / ( h ) + ∇ (2) ∆ − / ( h ) + R F ( h ) . The Heisenberg group H ν is noncommutative, with a more complicated geometry and Fourier trans-formation than the product Euclidean setting R m × R n of previous results [28] on flag Hardy spaces.To overcome these obstacles, we have created some new tools and techniques, which we suggest maybe helpful in solving related problems on the Heisenberg groups or more general stratified groups. • We define the Heisenberg group in an unusual way. In dealing with classical Hardy spaces, thebasic geometry is determined by cubes rather than by euclidean balls, and in our situation a similargeometry is appropriate. We use an ℓ ∞ rather than an ℓ distance to achieve this. To obtain a truedistance, rather than a quasidistance, we need an unusual parametrisation of the group structure. • In classical harmonic analysis on R n , dyadic decompositions play an important role; one keyfeature of these is that each dyadic cube may be written as a disjoint union of 2 n congruent cubes,each similar to the parent cube. In product harmonic analysis on R m × R n , an analogous role is playedby dyadic rectangles. In particular, in product Hardy space, maximal dyadic subrectangles of opensets are used to index the particles that make up an atom. There is a similar structure on H ν , asobserved by Strichartz [61], but the sets involved are rather irregular; further, there are nilpotent Lie10roups where such a structure cannot exist. To emphasize the analogy with the product space case,we defined atoms which are sums of particles associated to maximal adapted of an open set, but wealso show that it would suffice to consider particles indexed by tubes. Moreover, we clarify the sensein which an atom is a sum of particles, by requiring unconditional convergence in L ( H ν ). • In dealing with the geometry of the flag structure, we have to deal simultaneously with severalmetrics. For example, we deal with two heat kernels: one on the Heisenberg group, which may beestimated in terms of the Kor´anyi metric, and another in just the central variable, which involves aEuclidean metric. This leads to various complications, such as decompositions into annuli that involveboth metrics. To characterise the boundedness of singular integral on an individual atom, we use thetranslation and dilation on H ν to simply and reduce the estimate to the case of a particle a R whichis supported in a rectangle R centred at the origin of H ν and of width 1. Then the decomposition of H ν into annuli is straightforward. However, a direct decomposition of H ν into annuli for an arbitraryrectangle R is also feasible by combining the Euclidean metric on C ν and the Kor´anyi metric on H ν .This allows us to handle singular integrals that are not convolutions and further development on flagHardy spaces associated with more general operators. We discuss this in more depth in Section 8.1. • It is tricky to show the nontangential maximal function dominates the Lusin area function.In the multi-parameter tensor product setting R m × R n , the known tool to prove such a result isMerryfield’s lemma [45]. The key point of [45] is a “Cauchy–Riemann type” equation: for every even ϕ ∈ C ∞ ( R ) such that R ϕ ( x ) dx = 1, there is ψ ∈ C ∞ ( R ) with the same support as ϕ , such that R ψ ( x ) dx = 0 and ∂ t ϕ t ( x ) = ∂ x ψ t ( x ), where ϕ t ( x ) = t − ϕ ( x/t ) and ψ t ( x ) = t − ψ ( x/t ). However, onthe Heisenberg group and in more general contexts, it is not clear whether such a pair of functionsexists. Our approach bypasses the use of this construction and of Fourier transforms, and hence itmay be used in more general settings such as stratified Lie groups. We point out that even back tothe tensor product setting, our method is new. • The standard proof of the characterisation of Hardy space via Riesz transforms uses the radialmaximal function and a Cauchy–Riemann type equation. However, as before, the availability ofa Cauchy–Riemann type equation is unclear in our setting. Our new method dominates the flagLittlewood–Paley square function by the flag Riesz transform, by combining the singular integralcharacterisation of Christ and Geller with a randomisation argument, based on the Khinchin inequality.We expect that our method can be applied to the study of singular integral characterisation of differentversion of Hardy spaces in various settings. • In proving our results on Marcinkiewicz type multipliers, the fundamental tools that we use arethe atomic decomposition and the auxiliary weight w εj,ℓ : H ν → R + introduced by M¨uller, Ricci andStein [47]: w εj,ℓ ( z, t ) := 2 − ν ( j + ℓ ) (1 + 2 j + ℓ | z | ) ν (1+ ε ) − ℓ (1 + 2 ℓ | t | ) ε ∀ ( z, t ) ∈ H ν . We need to see the interaction of w εj,ℓ with one of the particles a R associated to an adapted rectangle R . The difficulty here is that the decomposition of H ν into rectangles is with respect to the Kor´anyimetric of H ν while the weight w εj,ℓ is described in terms of the Euclidean metrics in C ν and R (i.e., | z | and | t | ). In [47], they handled this via using a maximal function involved iterated one-dimensionalmaximal functions; this is bounded on L p ( H ν ) but not on our Hardy space. We use more refineddecompositions on H ν to overcome this difficulty.11 .5 Plan of the paper In Section 2, we discuss the geometry of the Heisenberg group, and prove or summarise some pre-liminary results. In Section 3, we examine the definitions of the atomic Hardy space in considerabledetail, and then show that the (usually sublinear) operators that define the other Hardy spaces are allbounded on the atomic space, thereby proving one half of many of the equivalences of Theorem 1.1. InSection 4, we examine the definition of the area function Hardy space in more detail and complete theproof that H F, area ( H ν ) = H F, atom ( H ν ). In Section 5, we examine the definition of the square functionHardy spaces in more detail and complete the proof that H F, cts ( H ν ) = H F, dis ( H ν ) = H F, atom ( H ν ). InSection 6, we examine the various maximal function Hardy spaces. It is clear, or we have alreadyshown that H F, atom ( H ν ) ⊆ H F, gmax ( H ν ) ⊆ H F, nontan ( H ν ) ⊆ H F, radial ( H ν );we show that H F, nontan ( H ν ) ⊆ H F, area ( H ν ), and outline why H F, nontan ( H ν ) = H F, radial ( H ν ); this lastfact is similar to the corresponding proof in the product-space setting, which may be found in [28,Subsection 3.2]. We complete our proof of Theorem 1.1 by showing that H F, area ( H ν ) = H F, Riesz ( H ν )in Section 7. Applications (and their proofs) will be provided in Section 8. In this section, we summarise relevant facts on the Heisenberg group and its geometry. The center of H ν is { (0 , t ) : t ∈ R } , and the canonical projection P of H ν onto C ν “forgets” the t -variable. We remind the reader that convolution is given by the formula f ∗ (1) f ( g ) = Z H ν f ( g ) f ( g − g ) dg = Z H ν f ( gg ) f ( g − ) dg ∀ g ∈ H ν . We also convolve a function f on H ν with a function f on R as follows: f ∗ (2) f ( g ) = Z R f ( gg ) f ( g − ) dg ∀ g ∈ H ν . For reasonable input functions, convolution is associative but not necessarily commutative. However,it is commutative on the space of radial functions , by which we mean the subspace of functions thatare radial in the z variable (see [38]) and on the space of even functions , that is, functions such that f ( g − ) = f ( g ) for all g ∈ H ν . If f and f both have compact support, then supp( f ∗ (1) f ) ⊆ supp( f ) · supp( f ).We will say that a function f on R is radial if it is even.Observe that, for suitable functions f and f on H ν and f on R , f ∗ (1) f ∗ (2) f ( g ) = Z H ν Z R f ( gg ) f ( g − g ) f ( g − ) dg dg = Z H ν Z R f ( gg g ) f ( g − ) f ( g − ) dg dg = Z H ν Z R f ( gg g ) f ( g − ) f ( g − ) dg dg , g is central. If f and f are even functions, then we may write f ∗ (1) f ∗ (2) f ( g ) = Z H ν Z R f ( gg g ) f ( g ) f ( g ) dg dg ∀ g ∈ H ν . (2.1)r The standard automorphic dilations δ r , where r >
0, on H ν are given by δ r ( z, t ) := ( rz, r t ). Forevery measurable subset E of H ν and every r > | δ r ( E ) | = r D | E | ; here, | E | denotes the measureof a subset E of H ν and D stands for the homogeneous dimension ν + 2 on H ν . Rotations in U( n ),acting on the right in the z variable and leaving t fixed, are also automorphisms of H ν .We identify vector fields with the associated first order differential operators. The Lie algebra ofleft-invariant vector fields on H ν is spanned by the following fields: X j = ∂∂x j + 4 νy j ∂∂t , Y j = ∂∂y j − νx j ∂∂t , and T = ∂∂t , (2.2)where j = 1 , . . . , ν . We write X n + j = Y j when j = 1 , . . . , ν . The vector fields X , . . . , X ν are calledhorizontal, while T is called vertical. We define the sub-Laplacian ∆ (1) on H ν to be − P νj =1 X j andthe Laplacian ∆ (2) to be −T ; the latter only involves the central variable. In the coordinate system( x + iy, t ), ∆ (1) = ν X j =1 ∂ ∂x j + ∂ ∂y j ! + 16 ν (cid:0) | x | + | y | (cid:1) ∂ ∂t , and ∆ (1) and ∆ (2) are even distributions.Note that X j is homogeneous of degree 1 with respect to the dilations δ r when 1 ≤ j ≤ ν , while T and ∆ (1) are homogeneous of degree 2, and ∆ (2) is homogeneous of degree 4, in the sense that X j ( f ◦ δ r ) = r ( X j f ) ◦ δ r , ∆ (1) ( f ◦ δ r ) = r (∆ (1) f ) ◦ δ r , T ( f ◦ δ r ) = r ( T f ) ◦ δ r , ∆ (2) ( f ◦ δ r ) = r (∆ (2) f ) ◦ δ r , when 1 ≤ j ≤ ν , r ∈ R + and f ∈ C ( H ν ). We write ∇ (1) and ∇ (2) for the horizontal and verticalgradients, that is, ( X , . . . , X ν ) and T . Finally, X j f may be considered as the convolution f ∗ (1) Υ j for an appropriate distribution Υ j with support at the origin; a similar conclusion holds for T f .The space of Schwartz functions on H ν is written S ( H ν ). This is the space of functions f such that p D f vanishes at infinity for all polynomials p on H ν and all left-invariant differential operators D .In the context of classical Hardy spaces, it is natural to focus on cubes rather than balls, and ananalogous comment applies in our setting. There are various left-invariant distances in use on H ν ; themost common are the control (or Carnot–Carath´eodory) distance d c , and the Kor´anyi distance d K .We shall use the gauge distance d , which is defined by setting d ( g, g ′ ) = (cid:13)(cid:13) g ′− · g (cid:13)(cid:13) = (cid:13)(cid:13) g − · g ′ (cid:13)(cid:13) ∀ g, g ′ ∈ H ν , (2.3)where k · k is given by k ( z, t ) k := max n | x | , | y | , . . . , | x ν | , | y ν | , | t | / o ∀ ( z, t ) ∈ H ν . (2.4)See [62, Section 2.2] for a discussion, and note that we have defined the group multiplication in anonstandard way to ensure that we have a norm that coincides with the usual ℓ ∞ norm in R ν +1 . We13rite B (1) ( g, r ) for the ball in H ν with centre g and radius r constructed using the distance d . We alsouse balls in the centre of H ν , which may be identified with R : we define B (2) ( t, s ) = { t ′ ∈ R : | t − t ′ |
Kor´anyi distance d K , given by d K ( g, g ′ ) = (cid:13)(cid:13) g ′− · g (cid:13)(cid:13) K = (cid:13)(cid:13) g − · g ′ (cid:13)(cid:13) K ∀ g, g ′ ∈ H ν , (2.6)where the Kor´anyi norm k · k K (with our definitions) is given by k ( z, t ) k K := ( k z k + 4 ν t ) / ∀ ( z, t ) ∈ H ν . (2.7)We do not need much about all these distances on H ν , but we mention that1 √ π d K ≤ d c ≤ d K and d ≤ d K ≤ (4 ν + 2 ν ) / d. (2.8)For the first pair of inequalities, see, for instance, [12, p. 140]; the second pair are elementary.Because our vector fields and distances are left-invariant, it is necessary to use right convolutionswith, for example, the flag Poisson kernel in the definition of the nontangential and radial maximalfunctions. For us, the flag Poisson kernel p r,s is the convolution kernel corresponding to the operator e r √ ∆ (1) e s √ ∆ (2) , which is a right convolution, that is, e r √ ∆ (1) e s √ ∆ (2) f = f ∗ (1) p r,s , where r, s ∈ R + . This creates a small but important difference between our work and that of Han,Lu and Sawyer [32], who used left convolutions. Reflecting functions (that is, composing with theinversion) exchanges left and right convolutions, so that in the end, the differences are minor, and wemay relate our Hardy spaces to theirs by such a composition.14 .2 Rectangles and Journ´e’s covering lemma Following [32], we use the work of [61, 62] on self-similar tilings to find a “nice” decomposition of H ν ,analogous to the decomposition of R n into dyadic cubes in classical harmonic analysis, and describe ananalogue of a lemma of Journ´e [39]. We identify C ν with R ν , | z | ∞ denotes max {| x | , | y | , . . . | x ν | , | y ν |} , τ denotes the cube [ − / , / ν , and H ν Z denotes the subgroup { ( z, t ) ∈ H ν : z ∈ Z ν , t ∈ (2 ν ) − Z } . Theorem 2.1 ([61, 62]) . There is a measurable function f : Q → R such that f (0) = 1 / and ν ( ν + 1) ≤ f ( z ) ≤ ν + 14 ν ( ν + 1) ∀ z ∈ Q , such that the set T o , defined to be T o := (cid:26) ( z, t ) : z ∈ Q , f ( z ) − ν ≤ t < f ( z ) (cid:27) , has the property that δ ν +1 ( T o ) = G g ∈ D g · T o , where D := { ( z, t ) ∈ H ν Z : | z | ∞ ≤ ν : | t | ≤ ν + 1 } . The definitions of T o and the metrics that we use show that T o ⊂ { ( z, t ) ∈ H ν : | z | ∞ ≤ / , | t | ≤ / } ⊆ ¯ B (1) ( o, / · ¯ B (2) (0 , /
8) = ¯ T ( o, / , / , (2.9)where the barred symbols indicate closures. We note that | T o | = 1 / ν while | T ( o, / , / | = 3 / Definition 2.2.
We define T = { g · T o : g ∈ H ν Z } , T j = δ (2 ν +1) j T and T := G j ∈ Z T j . We call the sets T ∈ T tiles . If j ∈ Z and g ∈ H ν Z and T = δ (2 ν +1) j ( g · T o ), then T = δ (2 ν +1) j ( g ) · δ (2 ν +1) j ( T o ), and we further definecent( T ) = δ (2 ν +1) j ( g ) , width( T ) = (2 ν + 1) j and height( T ) = (2 ν + 1) j ν . Theorem 2.3 ([32, 62, 61]) . Let T j and T be defined as above. Then the following hold1. for each j ∈ Z , T j is a partition of H ν , that is, H ν = F T ∈ T j T ;2. T is nested, that is, if T, T ′ ∈ T , then either T and T ′ are disjoint or one is a subset of the other;3. B (1) ( g, C q ) ⊆ T ⊆ B (1) ( g, C q ) , where g = cent( T ) and q = width( T ) for each T ∈ T ; theconstants C and C depend only on ν ;4. if T ∈ T j , then g · T ∈ T j for all g ∈ δ (2 ν +1) j H ν Z , and δ (2 ν +1) k T ∈ T j + k for all k ∈ Z . Every tile is a dilate and translate of the basic tile T o , so all have similar geometry. Hence eachtile in T j is a fractal set—its boundary is a set of Lebesgue measure 0 and (euclidean Hausdorff)dimension 2 ν —and is “approximately” a Heisenberg ball of radius (2 ν + 1) j . The decompositions are product-like in the sense that the tiles project onto cubes in the factor C ν , and their centers form a15roduct set. If two tiles in T j are “horizontal neighbours”, then the distance between their centres is(2 ν + 1) j , while if they are “vertical neighbours”, then the distance is (2 ν + 1) j / ν .Han, Lu and Sawyer [32] used unions of tiles to pursue the analogy with rectangles in the plane R . We follow them, but with different nomenclature to reflect the fact that our objects are not basedon powers of 2. Given tiles T and T ′ , such that T ⊂ T ′ , the projection P ( T ) of T onto C ν is a cube, Q say; let R = P − ( Q ) ∩ T ′ . Then R is the (finite) union of the tiles T ′′ ∈ T j such that T ′′ ⊂ T ′ and P ( T ′′ ) = Q . Definition 2.4.
The adapted rectangle determined by tiles T and T ′ such that T ⊂ T ′ is definedto be the set P − ( P ( T )) ∩ T ′ . The centre of R , written cent( R ), is the centre of the interval P − (cent( P ( T ))) ∩ T ′ , where cent( Q ) is the centre of Q ; the width of R , written width( R ), is width( T )and the height of R , written height( R ), is height( T ′ ). The collection of all adapted rectangles is de-noted R .We note that the collection of all adapted rectangles is countable. Adapted rectangles are calledvertical dyadic rectangles by Han, Lu and Sawyer [32]; they also defined horizonal dyadic rectangles,but we do not need to deal with these.When we consider atomic Hardy spaces, we will want to consider functions supported in open sets,which may be written as sum of functions supported in enlargements of maximal adapted rectanglesof the open set. Recall that the tube T ( g, r, s ) is the set g · B (1) ( o, r ) · B (2) (0 , s ); evidently, | T ( g, r, s ) | = 2 ν +1 r ν ( r + s ) . (2.10) Definition 2.5.
Fix κ ∈ [1 , ∞ ). We define the enlargement R ∗ ,κ , usually written R ∗ , of an adaptedrectangle R to be the tube T ( g, κq/ , κ (4 h + q ) / g = cent( R ), q = width( R ), and h =height( R ). Lemma 2.6.
For all R ∈ R and κ ∈ (1 , ∞ ) , R ⊂ R ∗ . (2.11) Further, when κ ≥ (2 ν + 1) , given any tube T , there exists an adapted rectangle R such that R ⊆ T ⊆ R ∗ ,κ , (2.12) Proof.
By part (4) of Theorem 2.3, (2.11) is invariant under certain dilations and translations. Hencewe may assume that cent( R ) = o , width( R ) = 1, and height( R ) = (2 ν + 1) j / ν for some j ∈ N . Inthis case, R is made up of (2 ν + 1) j copies of T o stacked vertically on each other; more precisely, R = G | k |≤ h/ (0 , k ) · T o ⊆ T o · ¯ B (2) (0 , h/ ⊆ ¯ B (1) ( o, / · ¯ B (2) (0 , h/ / k ∈ Z / ν in the union. Now (2.11) follows easily.The assertion (2.12) is the H ν version of the fact that every interval [ a, b ) in R contains an intervalwith centre (2 ν + 1) j k and width (2 ν + 1) j such that [ a, b ) is contained in the interval with the samecentre and width(2 ν + 1) j +2 ; it suffices to take j = ⌊ log ν +1 ( b − a ) ⌋ − k = ⌈ log ν +1 a ⌉ + 1.16or future reference, we note that, if R is an adapted rectangle, cent( R ) = g , width( R ) = q andheight( R ) = h , then R ⊂ ¯ T ( g, q/ , (4 h + q ) / , | R | = q ν h,R ∗ = T ( g, κq/ , κ (4 h + q ) / , | R ∗ | = κ D q ν ( h + q / , ≤ | R ∗ | κ D | R | ≤ ν/ h ≥ q / ν ). Controlling averages over adapted rectangles is essentiallythe same as controlling averages over tubes, and we use a maximal operator to do this. Our “flag”maximal operator is “bigger” than the usual Hardy–Littlewood maximal operator, but “smaller” thanthe “strong” maximal operator used in [46, 47]. Recall that χ (1) r and χ (2) s denote the normalisedcharacteristic functions of the unit ball in H ν and the unit interval in the central variable; χ r,s is shortfor χ (1) r ∗ (2) χ (2) s . Definition 2.7.
We define the flag maximal operator M F by M s ( f )( g ) = sup r,s ∈ R + | T ( g, r, s ) | Z T ( g,r,s ) | f ( g ′ ) | dg ′ ∀ g ∈ H ν . (2.14)and the iterated maximal operator M it by M it ( f )( g ) = sup r,s ∈ R + | f | ∗ (1) χ r,s ( g ) ∀ g ∈ H ν . (2.15) Lemma 2.8.
Suppose that f is a measurable function on H ν . Then | f | ∗ (1) χ r,s/ ( g ) . | T ( g, r, s ) | Z T ( g,r,s ) | f ( g ′ ) | dg ′ . | f | ∗ (1) χ r,s ( g ) ∀ g ∈ H ν . Consequently M s ( f ) h M it ( f ) . Both maximal operators are bounded on L p ( H ν ) when < p ≤ ∞ .Proof. We write g ∈ H ν as ( z, t ). By definition,1 | T ( g, r, s ) | Z T ( g,r,s ) (cid:12)(cid:12) f ( g ′ ) (cid:12)(cid:12) dg ′ = Z Z C ν × R (cid:12)(cid:12) f (( z, t ) · ( z ′ , t ′ )) (cid:12)(cid:12) v ( r,s ) ( z ′ , t ′ ) dt ′ dz ′ , (2.16)where v ( r,s ) ( z ′ , t ′ ) = 1 | B (1) ( o, r ) · B (2) (0 , s ) | Q (0 ,r ) ( z ′ ) B (2) (0 ,r + s ) ( t ′ ) , and similarly | f | ∗ (1) χ (1) r ∗ (2) χ (2) s )( g ) = Z B (1) ( o,r ) Z B (2) (0 ,s ) (cid:12)(cid:12) B (1) ( o, r ) (cid:12)(cid:12) (cid:12)(cid:12) B (2) (0 , s ) (cid:12)(cid:12) (cid:12)(cid:12) f ( g · g ′ · g ′′ ) (cid:12)(cid:12) dg ′′ dg ′ = Z Z C ν × R (cid:12)(cid:12) f (( z, t ) · ( z ′ , t ′ )) (cid:12)(cid:12) w ( r,s ) ( t ′ ) dt ′ dz ′ , (2.17)and w ( r,s ) ( t ′ ) = 1 (cid:12)(cid:12) B (1) ( o, r ) (cid:12)(cid:12) (cid:12)(cid:12) B (2) (0 , s ) (cid:12)(cid:12) Q (0 ,r ) ( z ′ ) Z R B (2) (0 ,r ) ( t ′ + t ′′ ) B (2) (0 ,s ) ( t ′′ ) dt ′′ . Nowmin { r , s } B (2) (0 , ( r + s ) / ( t ′ ) ≤ Z R B (2) (0 ,r ) ( t ′ + t ′′ ) B (2) (0 ,s ) ( t ′′ ) dt ′′ ≤ { r , s } B (2) (0 ,r + s ) ( t ′ )17nd min { r , s } (cid:12)(cid:12) B (1) ( o, r ) (cid:12)(cid:12) (cid:12)(cid:12) B (2) (0 , s ) (cid:12)(cid:12) h min { r , s } r D s = 1max { r , s } r ν h r + s ) r ν h (cid:12)(cid:12) B (1) ( o, r ) · B (2) (0 , s ) (cid:12)(cid:12) , whence v ( r,s/ . w ( r,s ) . v ( r,s ) , and substituting these inequalities into (2.16) and (2.17), and thentaking suprema, completes the proof of the equivalence of the two maximal functions.Since | f |∗ (1) χ (1) r ∗ (2) χ (2) s = | f |∗ (2) χ (2) s ∗ (1) χ (1) r , the maximal operator M it may be interpreted as thecomposition of a Hardy–Littlewood maximal function in the central variable with a Hardy–Littlewoodmaximal function on H ν , in either order. It is evident that it is bounded on L p ( H ν ) when 1 < p ≤ ∞ ,and it is certainly unbounded on L ( H ν ). Lemma 2.9.
Let R be an adapted rectangle. Then M F R ( g ′ ) ≥ ν + 2) − κ − D , ∀ g ′ ∈ R ∗ ,κ . Proof.
Write g = cent( R ), q = width( R ), and h = height( R ). Suppose that g ′ , g ′′ ∈ R ∗ . By definition, g ′ ∈ g · B (1) ( o, κq/ · B (2) (0 , κ ( q + 4 h ) / g ∈ g ′ · B (1) ( o, κq/ · B (2) (0 , κ ( q + 4 h ) / g ′′ ∈ g · B (1) ( o, κq/ · B (2) (0 , κ ( q + 4 h ) / ⊆ g ′ · B (1) ( o, κq/ · B (2) (0 , κ ( q + 4 h/ · B (1) ( o, κq/ · B (2) (0 , κ ( q + 4 h ) / g ′ · B (1) ( o, κq ) · B (2) (0 , κ ( q + 4 h ) / . Hence R ⊆ R ∗ ⊆ g ′ · B (1) ( o, κq ) · B (2) (0 , κ ( q + 4 h ) / M F R ( g ′ ) ≥ | B (1) ( o, κq ) · B (2) (0 , κ ( q + 4 h ) / | Z B (1) ( g ′ ,κq ) · B (2) (0 ,κ ( q +4 h ) / R ( g ′′ ) dg ′′ = | R || B (1) ( o, κq ) · B (2) (0 , κ ( q + 4 h ) / | = q ν h ν +1 ( κq ) ν κ (5 q + 4 h ) / ≥ ν κ D (5 ν + 2) , since h ≥ q / ν . Definition 2.10.
Suppose that Ω is an open set of H ν of finite measure | Ω | . We write R (Ω) for thecollection of all R in R whose interior is a subset of Ω, and M (Ω) for the collection of all maximalsuch R in R (Ω).We abuse language a little and call rectangles R ∈ R (Ω) adapted subrectangles of Ω. Each suchsubrectangle R has an enlargement R ∗ , and we can control the measure of S R ∈ R (Ω) R ∗ . Corollary 2.11.
Suppose that Ω is an open subset of H ν of finite measure. Then (cid:12)(cid:12)(cid:12)(cid:12) [ R ∈ R (Ω) R ∗ (cid:12)(cid:12)(cid:12)(cid:12) . | Ω | . (2.18) The implicit constant depends only on the enlargement parameter κ and ν .Proof. By definition, M F Ω ( g ′ ) ≥ M F R ( g ′ ) ≥ ν + 2) − κ − D , say, by Lemma 2.9. Since the flagmaximal function is bounded on L ( H ν ), |{ g ∈ H ν : |M s Ω ( g ) | ≥ λ }| . k Ω k λ ∀ λ ∈ R + , where the implicit constant depends only on ν , and taking λ to be 2 / (5 ν + 2) shows (2.18).18or an adapted rectangle R , there is a unique adapted rectangle, R † say, that contains R and is atranslate of δ ν +1 R . It is easy to check that, if the enlargement parameter κ in the definition of R ∗ islarge enough, then R † ⊆ R ∗ . Corollary 2.12.
Suppose that Ω is an open subset of H ν of finite measure. Then (cid:12)(cid:12)S R ∈ M (Ω) R † (cid:12)(cid:12) . | Ω | . The implicit constant depends only on ν . We now recall Journ´e’s covering lemma, which was first proved by Journ´e [39] in R × R , and laterby Pipher [53] in higher dimensions. It has been extended to products of spaces of homogeneous type;see, for example [29]. In the flag setting, Journ´e’s covering lemma is actually a special case of theresult for products of spaces of homogeneous type. We refer to [32, Lemma 14] for the statement ofthis special version of the covering lemma.Suppose that Ω is an open subset of H ν of finite measure, and fix α ∈ (0 , (cid:26) g ∈ H ν : M F ( χ Ω )( g ) > α (cid:27) . For any R ∈ M (Ω), there may be several maximal R ′ ∈ M ( ˜Ω) that contain R . However, for a givenbase side-length, there is a unique such rectangle; indeed, if R ⊆ R ′ ⊆ ˜Ω and R ⊆ R ′′ ⊆ ˜Ω, and R ′ and R ′′ have the same base side-length, then the projections P R ′ and P R ′′ of R ′ and R ′′ onto C ν are(2 ν +1)-adic cubes of the same size that contain P R , so they must coincide, and then R ⊆ R ′ ∪ R ′′ ⊆ ˜Ω;if R ′ and R ′′ are both maximal, then R ′ = R ′′ .Here is Journ´e’s lemma, in a form in which we will use it. Lemma 2.13.
Suppose that Ω is an open subset of H ν of finite measure. For any δ ∈ R + , X R ∈ M (Ω) (cid:18) width( R )width( R ↑ ) (cid:19) δ | R | ≤ c δ | Ω | and X R ∈ M (Ω) (cid:18) height( R )height( R ↑ ) (cid:19) δ | R | ≤ c δ | Ω | . where c δ does not depend on Ω . Evidently c δ depends on δ , but it also depends on the parameter α in the definition of ˜Ω. Wesay that a constant is geometric if it depends on inherent properties of the Heisenberg group andits geometry (including its decompositions into tiles and rectangles), the apertures of the cones thatappear in the various definitions, and the enlargement factors that connect adapted rectangles andsupports of particles; a geometric multiple is defined similarly. The constants mentioned above aregeometric, except that some depend on p , others on M and N , and others on positive parameters α and δ . We use the notation A . B to mean that there is a constant C such that A ≤ CB , and A h B to mean that A . B and A . B . If the constant is geometric, we do not necessarily point this outexplicitly. However, when the implicit constant in one of these inequalities depends on a nongeometricconstant, we indicate this explicitly, for example, we might write P k ∈ N − δk h δ
1. The constant α that we use in the definition of ˜Ω will be geometric; so will δ in Lemma 2.13. On the Heisenberg group, radial functions (and radial distributions), that is, those that are invariantunder the action of the rotations in U( n ), form a commutative algebra. This was noticed in [38]19though as is often the case with analysis on H ν , the necessary calculations to show this may be foundin [24], but the result is not stated explicitly). In particular the differential operators ∆ (1) and i T areradial and essentially self-adjoint. Spectral theory, combining the general features of abstract theory,as in, for instance, [44], with the additional features available because we are working on a Lie group,has been used to develop a functional calculus for these operators (see [2] and the references citedthere), so that expressions Φ(∆ (1) , i T ) and Φ(∆ (1) , ∆ (2) ) are defined for functions Φ in S ( R ), andthese operators are given by right convolution with kernels k Φ in S ( H ν ). The functional calculus maybe extended to other classes of functions, such as bounded continuous functions, or rational functions,at the cost of using distributional kernels.Let F denote the Heisenberg fan, that is, { ((2 d + n ) | λ | , λ ) ∈ R : d ∈ N , λ ∈ R \ { }} ∪ { ( ξ ,
0) : ξ ∈ R + } . We shall use the following features of this theory.
Theorem 2.14.
There exists an isometry F : L ( H ν ) → L d ∈ N L ( R ; H d ) , where each H d is a Hilbertspace, such that1. k f k L ( H ν ) = X d ∈ N Z R kF f ( d, λ ) k H d λ D − dλ ! / for all f ∈ L ( H ν ) ;2. F (Φ(∆ (1) , i T ) f )( d, λ ) = Φ((2 d + n ) | λ | , λ ) F ( f )( d, λ ) for all f ∈ Dom Φ(∆ (1) , i T ) , all d ∈ N , all λ ∈ R , and all Schwartz functions or polynomials Φ ;3. let A be a continuous linear operator from S ( H ν ) to its dual space S ′ ( H ν ) that commutes with(left) translations. Then A has a convolution kernel k Φ in S ( H ν ) , which is necessarily radial, ifand only if there exists Φ ∈ S ( R ) such that A = Φ(∆ (1) , i T ) . If Φ : R → R is even in the second variable, we may interpret the theorem as giving informationabout Ψ(∆ (1) , ∆ (2) ), where Ψ( µ, λ ) = Φ( µ, λ ).In particular, the space S ∆ (1) ( H ν ) of kernels of convolution operators Φ(∆ (1) ), where Φ ∈ S ( R ),is a subalgebra of the convolution algebra of Schwartz functions on H ν , and the space S ∆ (2) ( R ) ofkernels of convolution operators Φ(∆ (1) ), where Φ ∈ S ( R ), is an algebra that may be identified withthe convolution algebra of even Schwartz functions on the centre R of H ν . Given a kernel ϕ in one ofthese algebras, we denote by G ϕ the Gel’fand transform of ϕ , that is, the function m such that ϕ isthe kernel of m (∆ (1) ) or m (∆ (2) ).With a view to discussing the possibility of generalisation of our results later, we mention thatthis result has been extended to more general contexts, including stratified nilpotent groups, byMartini [43]. Corollary 2.15.
Suppose that
Ψ : R + × R + → C is bounded and continuous, and that C (Ψ) := Z Z R + × R + | Ψ( r, s ) | dss drr < ∞ . Then (cid:18)Z R + × R + (cid:13)(cid:13) Ψ( r ∆ (1) , s ∆ (2) ) f (cid:13)(cid:13) L ( H ν ) dss drr (cid:19) / = C (Ψ) k f k L ( H ν ) . (2.19)20 roof. Observe that (cid:18)Z R + × R + (cid:13)(cid:13) Ψ( r ∆ (1) , s ∆ (2) ) f (cid:13)(cid:13) L ( H ν ) dss drr (cid:19) / = Z Z R + × R + X d ∈ N Z R (cid:13)(cid:13) F (cid:0) Ψ( r ∆ (1) , s ∆ (2) ) f (cid:1) ( λ, d ) (cid:13)(cid:13) H d λ D − dλ dss drr ! / = Z Z R + × R + X d ∈ N Z R | Ψ( r (2 d + 1) | λ | , sλ ) | kF f ( λ, d ) k H d λ D − dλ dss drr ! / = X d ∈ N Z R Z Z R + × R + | Ψ( r (2 d + 1) | λ | , sλ ) | dss drr kF f ( λ, d ) k H d λ D − dλ ! / = (cid:18)Z Z R + × R + | Ψ( r, s ) | dss drr (cid:19) / X d ∈ N Z R kF f ( λ, d ) k H d λ D − dλ ! / = (cid:18)Z Z R + × R + | Ψ( r, s ) | dss drr (cid:19) / k f k L ( H ν ) , as required.This proof is a very mild generalisation of the analogous result for one-variable functional calculus,which is well known. See, for example, [1, p. 101]; see also the reproducing formula in one-parameterin [34, equation (4.21)]). We may prove the following reproducing formula similarly; we leave thedetails to the reader. Corollary 2.16.
Suppose that Ψ (1) : R + → R and Ψ (2) : R + → R are bounded nonnegative continuousfunctions, and that Z R + Ψ (1) ( r ) drr = 1 and Z R + Ψ (2) ( s ) dss = 1 . Then f = Z Z R + × R + (cid:16) Ψ (1) ( r ∆ (1) )Ψ (2) ( s ∆ (2) ) (cid:17) f dss drr for all f ∈ L ( H ν ) . The requirements on Ψ (1) and Ψ (2) may be satisfied by taking nonnegative bounded continuousfunctions that vanish sufficiently rapidly at 0 and at ∞ , observing that the integrals in question areboth finite, and then normalising the functions so that the values of the integrals both become 1. Thenonnegativity restriction on the functions may be lifted provided that Z ∞ (cid:12)(cid:12)(cid:12) Ψ (1) ( r ) (cid:12)(cid:12)(cid:12) drr < ∞ and Z (cid:12)(cid:12)(cid:12) Ψ (2) ( s ) (cid:12)(cid:12)(cid:12) dss < ∞ . Remark . We observe that, if supp Φ (1) ⊆ (0 , a ] and supp Φ (2) ⊆ [ b, ∞ ), where 0 < a ≤ n √ b < ∞ ,then Ψ (1) ( r ∆ (1) )Ψ (2) ( s ∆ (2) ) f = 0 , ∀ f ∈ L ( H ν ) , whenever r > √ s . These conditions ensure thatΨ (1) ( r (( n + 2 d ) | λ | ))Ψ (2) ( sλ ) = 0 for all d ∈ N and all λ ∈ R .We choose even S ( R )-functions Φ (1) and Φ (2) whose Fourier transforms are supported in a smallneighbourhood of 0 in R , and define Ψ (1) and Ψ (2) to be the Schwartz functions x x M Φ (1) ( x )and x x M Φ (2) ( x ). By [34, Lemma 3.5], once the supports of Φ (1) and Φ (2) are small enough,the supports of the convolution kernels of Φ (1) ( √ ∆ (1) ) and Φ (2) ( √ ∆ (2) ) are subsets of B (1) ( o,
1) and21 (2) (0 ,
1) respectively. Since Ψ (1) ( √ ∆ (1) ) = ∆ M (1) Φ (1) ( √ ∆ (1) ) and Ψ (2) ( √ ∆ (2) ) = ∆ M (2) Φ (2) ( √ ∆ (2) ), thesupports of the convolution kernels of these operators are also supported in B (1) ( o,
1) and B (2) (0 , w Z R + Ψ (1) ( µ ) e − wµ dµ and w Z R + Ψ (2) ( λ ) e − wλ dλ are holomorphic in { w ∈ C : Re( w ) > } and not identically 0 there. Hence there are many choices of w in R + so that the integrals do not vanish identically, and by rescaling Ψ (1) and Ψ (2) , we may firstsuppose that they do not vanish when w = 1, and then that the integrals are equal to 1. This impliesthe reproducing formula f = Z Z R + × R + Ψ (1) ( r √ ∆ (1) )Ψ (2) ( s √ ∆ (2) )( r √ ∆ (1) e − r √ ∆ (1) s √ ∆ (2) e − s √ ∆ (2) )( f ) drr dss for all f ∈ L ( H ν ). H ν and cancellation We say that a function f on H ν is homogeneous of degree d if f ( δ r g ) = r d f ( g ) and a differentialoperator D on H ν is homogeneous of degree e if D ( f ◦ δ r ) = r e D ( f ) ◦ δ r . If f is homogeneous ofdegree d and D is homogeneous of degree e , then the function D f is homogeneous of degree d − e .The coordinate functions z j (and their real and imaginary parts x j and y j ) are homogeneous of degreeone, while the coordinate t is homogeneous of degree 2, and each homogeneous polynomial p has apositive degree, which is the product of the degrees of the coordinate functions involved in each of themonomial terms of p . Lemma 2.18.
Let Φ ∈ S ( R ) , m be a positive integer, t > , and k be the convolution kernel of theoperator ( t ∆ (1) ) m Φ( t ∆ (1) ) . For all polynomials p of homogeneous degree less than m , Z H ν p ( g ) k ( g ) dg = 0 . (2.20) Proof.
Let k ′ be the convolution kernel of the operator Φ( t ∆ (1) ); then k = ( t ∆ (1) ) m k ′ , and byfunctional calculus, k ′ in S ( H ν ). An integration by parts shows that Z H ν p ( g ) k ( g ) dg = Z H ν p ( g )( t ∆ (1) ) m k ′ ( g ) dg = t m Z H ν (∆ m (1) p )( g ) k ′ ( g ) dg. Now ∆ m (1) p is a polynomial of negative degree, so it is 0, and the lemma holds.A closely related result concerns the cancellation property for atoms. Lemma 2.19.
Suppose that b is in Dom(∆ M (1) ∆ M (2) ) in L ( H ν ) and has compact support, and let a = Dom(∆ M (1) ∆ M (2) ) b . If p is a monomial on H ν , of degree d in z and degree d in t , where d ≤ M , d ≤ M and d + d < M , then R H ν p ( g ) a ( g ) dg = 0 . Proof.
The proof is almost identical to that of the previous lemma, and we leave the details to thereader. 22 .5 Estimates for the heat and Poisson kernels
Let h (1) r (where r >
0) be the heat kernel for ∆ (1) , that is, the convolution kernel of e − r ∆ (1) on H ν .By homogeneity, h (1) r ( z, t ) = r − D/ h (1)1 ( δ / √ r ( z, t )) = r − D/ h (1)1 ( z/ √ r, t/r ) ∀ r > ∀ ( z, t ) ∈ H ν . (2.21)The following Gaussian upper bound for the heat kernel in terms of the control norm k·k c holds: | ∆ m (1) ∆ n (2) h (1) r ( z, t ) | . m,n,δ r − m − n − D/ exp (cid:16) − k ( z, t ) k c δ ) r (cid:17) . (2.22)for all δ ∈ R + and all m, n ∈ N . This is proved in [63, p. 48]. There is a similar lower bound (see [63,p. 61]) for the heat kernel (but not its derivatives), namely, h (1) r ( z, t ) & δ r − D/ exp (cid:16) − k ( z, t ) k c − δ ) r (cid:17) . The subordination formula e − λ = 1 √ π Z R + e − v √ v e − λ / v dv ∀ λ ∈ R + leads us to corresponding estimates for the Poisson kernel p (1) . By functional calculus, we may write e − r √ ∆ (1) = 1 √ π Z R + e − v √ v e − r ∆ (1) / v dv, for all r ∈ R + , ∆ m (1) ∆ n (2) e − r √ ∆ (1) = 1 √ π Z R + e − v √ v ∆ m (1) ∆ n (2) e − r ∆ (1) / v dv and so from (2.22), (cid:12)(cid:12)(cid:12) ∆ m (1) ∆ n (2) p (1) r ( g ) (cid:12)(cid:12)(cid:12) . Z R + e − v √ v (cid:18) r v (cid:19) − m − n − D/ e − v k g k c / (1+ ε ) r dv = 2 m +4 n + D r − m − n − D Z R + v m +2 n + D/ − / e − v − v k g k c / (1+ ε ) r dv = 2 m +4 n + D r − m − n − D Γ( m + 2 n + D/ / k g k c / (1 + ε ) r ) m +2 n + D/ / h r ( r + k g k c ) m +2 n + D/ / , for all g ∈ H ν ; the implicit constants depend on m , n , ν and ε . Analogously, p (1) r ( g ) & r ( r + k g k c ) D/ / ∀ g ∈ G. For computation, we may and shall replace the control norm by the gauge norm, and we shall use theestimates p (1) r ( g ) h r ( r + k g k ) D/ / , (cid:12)(cid:12)(cid:12) ∆ m (1) ∆ n (2) p (1) r ( g ) (cid:12)(cid:12)(cid:12) . r ( r + k g k ) m +2 n + D/ / (2.23)for all g ∈ G . 23imilar estimates hold for the usual Poisson kernel p (2) s on R , namely, p (2) s ( t ) h ss + k t k , (cid:12)(cid:12)(cid:12) ∆ n (2) p (2) s ( t ) (cid:12)(cid:12)(cid:12) . s ( s + k t k ) n +1 (2.24)for all t ∈ R and Z R (cid:12)(cid:12)(cid:12) ∆ n (2) p (2) s ( t ) (cid:12)(cid:12)(cid:12) dt . s n ∀ s ∈ R + . The implicit constants are geometric.In various situations, we are going to deal with functions that behave much as the Poisson kernelson H ν and on R . Definition 2.20.
A family F ( H ν ) of pairs of functions ( ϕ (1) , ϕ (2) ), where ϕ (1) : H ν → C and ϕ (2) : R → C , is said to be Poisson-bounded if there are constants C m and C n for all m ∈ N and all n ∈ N such that (cid:12)(cid:12)(cid:12) D ϕ (1) ( g ) (cid:12)(cid:12)(cid:12) ≤ C m (1 + k g k ) m + D/ / ∀ g ∈ H ν (2.25)for all differential operators D that are products of m vector fields, each chosen from X , . . . , X ν , and (cid:12)(cid:12)(cid:12) T e ϕ (2) ( t ) (cid:12)(cid:12)(cid:12) ≤ C n (1 + k t k ) n +1 ∀ t ∈ R . (2.26)for all pairs ( ϕ (1) , ϕ (2) ) ∈ F ( H ν ). The family F ( H ν ) may consist of one pair ( ϕ (1) , ϕ (2) ) only; in thiscase we say that the pair ( ϕ (1) , ϕ (2) ) is Poisson-bounded.Given such a Poisson-bounded family, the grand maximal function M F, gmax ( f ) of f ∈ L ( H ν ) isdefined by M F, gmax ( f )( g ) = sup ϕ ∈ F ( H ν ) sup r,s ∈ R + (cid:12)(cid:12) f ∗ (1) ϕ r,s ( g ) (cid:12)(cid:12) ∀ g ∈ H ν . Clearly, the grand maximal function depends on the family F ( H ν ); moreover, in any calculationinvolving Poisson-bounded families of functions, it is only necessary to control finitely many derivatives,but the number of derivatives varies from one calculation to another, and depends on ν ; we do notbother keeping tabs on these dependencies.In the next lemma, we compare two maximal functions. We recall that χ r,s = χ (1) r ∗ (2) χ (2) s , where χ (1) r is the L ( H ν )-normalised characteristic function of the ball B (1) ( o, r ) and χ (2) s is the L ( R )-normalised characteristic function of the ball B (2) (0 , s ). Lemma 2.21.
For all β, γ ∈ R + , χ βr,γs . α,β p r,s . α,β P j,k ∈ N − j − k χ j βr, k γs , ∀ r, s ∈ R + . Hence M F, gmax ( | f | ) ≃ M s ( | f | ) for all measurable functions f on H ν .Proof. It follows immediately from (2.23) that χ (1) r . p (1) r . P j ∈ N − j p (1)2 j r χ (1) r and a similar estimateholds for p (2) s and χ (2) s . The lemma follows.In particular, the grand maximal operator is bounded on L p ( H ν ) when 1 < p ≤ ∞ . A similarresult is shown in [18]. 24 .6 Flag Riesz transforms and singular integrals We recall that the (tensor-valued) flag Riesz transformation is defined by R F = ∇ (1) ∆ − / ∇ (2) ∆ − / . It is well known that ∇ (2) ∆ − / is the Hilbert transform, which is a convolution with a singular kernel.Christ and Geller [8] show that ∇ (1) ∆ − / may also be considered as a convolution with a singularkernel (plus, perhaps, a Dirac delta distribution at the origin). In brief, the argument goes that ∆ − / is a convolution operator with a smooth kernel, homogeneous of degree 1 − D , and ∇ (1) ∆ − / is abounded operator on L ( H ν ) by spectral theory. Then ∇ (1) ∆ − / is a convolution operator with adistribution that is homogeneous of degree D , that may be identified with a smooth kernel away fromthe identity. The L boundedness then implies that the smooth kernel has “mean zero” in a suitablesense and convolution with this kernel may be interpreted in various limiting senses, such as a principalvalue; for more on this, see [22] and [40]. It also implies that any strictly distributional component of ∇ (1) ∆ − / is a multiple of the Dirac delta distribution at the group identity.A very similar argument shows that the space of convolution operators with distributions on H ν that are homogeneous of degree D and given by a smooth function away from the origin, and actboundedly on L ( H ν ) by convolution, forms an algebra.We do not go into this in any detail, but leave the reader to consult the cited references. Definition 2.22. A simple singular integral kernel on H ν (of type 0) is a smooth kernel k on H ν \ { o } ,homogeneous of degree − D , of mean 0 in the sense of Knapp and Stein [40]. A simple singular integraloperator is a convolution (on the left) by such a kernel.It may be shown that simple singular integral operators on H ν are bounded on L p ( H ν ) when1 < p < ∞ and on the Folland–Stein–Christ–Geller Hardy space H F SCG ( H ν ). In Section 8 weconsider more general flag singular integrals studied by Phong and Stein [54]. Because we are dealing with two Laplacians, there are various Sobolev-type inequalities possible. Wepresent several that will be useful for us.
Lemma 2.23.
Suppose that b ∈ L ( H ν ) is supported in a set R = { ( z, t ) ∈ H ν : z ∈ Q, f ( z ) ≤ t < f ( z ) + h } , for some cube Q in C ν of side-length q and measurable function f : Q → R . If ∆ (2) b ∈ L ( H ν ) , then (cid:18)Z R | b ( g ) | dg (cid:19) / . h (cid:18)Z R (cid:12)(cid:12) ∆ (2) b ( g ) (cid:12)(cid:12) dg (cid:19) / . Similarly, if ∆ (1) b ∈ L ( H ν ) , then (cid:18)Z R | b ( g ) | dg (cid:19) / . q (cid:18)Z R (cid:12)(cid:12) ∆ (1) b ( g ) (cid:12)(cid:12) dg (cid:19) / . The implicit constants are absolute in the first case, and depend on ν in the second and third.Proof. It is well-known, and very easy to check using Fourier sine series, that (cid:18)Z R | c ( t ) | dt (cid:19) / ≤ h π Z R (cid:12)(cid:12)(cid:12)(cid:12) d dt c ( t ) (cid:12)(cid:12)(cid:12)(cid:12) dt ! / c on R supported in an interval of length h . The first estimate is proved by integratingthe inequality above in z .To prove the second estimate, by translating and dilating, it suffices to suppose that Q is the cubewith centre 0 in C ν and side-length 1. By Fourier analysis in the t variable, any function b on R may beexpressed as an integral of functions b λ that satisfy the conditions b λ ( z, t ) = b λ ( z, e iλt , ∀ ( z, t ) ∈ H ν , and k b k L ( H ν ) = (cid:18)Z R k b λ ( · , k L ( C ν ) dλ (cid:19) / . Now ∆ (1) b λ ( z, t ) = − ν X j =1 ∂ ∂x j + 16 ν λ | z | b λ ( z, t ) ∀ ( z, t ) ∈ H ν . Considered as an operator on L ( Q ), the “truncated Hermite operator” − P νj =1 ∂ /∂x j + 16 ν λ | z | has discrete spectrum, with eigenfunctions ϕ , say. Since Z Q ¯ ϕ ( z ) − ν X j =1 ∂ ∂x j + 16 ν λ | z | ϕ ( z ) dz ≥ Z Q ¯ ϕ ( z ) − ν X j =1 ∂ ∂x j ϕ ( z ) dz, the eigenvalues of this operator are at least as big as those of the Euclidean Laplacian, which by aFourier sine series argument are at least 2 ν/π . It follows that (cid:13)(cid:13) ∆ (1) b (cid:13)(cid:13) L ( H ν ) = (cid:18)Z R (cid:13)(cid:13) ∆ (1) b λ ( · , (cid:13)(cid:13) L ( C ν ) dλ (cid:19) / ≥ (cid:18)Z R ν π k b λ ( · , k L ( C ν ) dλ (cid:19) / = 2 νπ k b k L ( H ν ) , as required. We are going to deal with various square functions, and here we prove that different square functionsare equivalent, under appropriate conditions.
Definition 2.24.
Suppose that ( ϕ (1) , ϕ (2) ) is a Poisson-bounded pair, as in Definition 2.20, and that R H ν ϕ (1) ( g ) p ( g ) dg = 0 for all polynomials p on H ν of homogeneous degree up to M with M > ν , andthat R R ϕ (2) ( s ) p ( s ) ds = 0 for all polynomials p on R of degree up to 1.For f ∈ L p ( H ν ), we define the continuous Littlewood–Paley square function S F, cts , ϕ ( f ) associatedto ϕ (1) and ϕ (2) by S F, cts , ϕ ( f )( g ) := (cid:18)Z Z R + × R + (cid:12)(cid:12) f ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s ( g ) (cid:12)(cid:12) drr dss (cid:19) / for all g ∈ H ν . Lemma 2.25.
Suppose that ( ϕ (1) , ϕ (2) ) and ( ψ (1) , ψ (2) ) satisfy the boundedness and cancellation con-ditions of in Definition 2.24. Then kS F, cts , ψ ( f ) k L ( H ν ) h kS F, cts , ϕ ( f ) k L ( H ν ) for all f ∈ L ( H ν ) . roof. First, we establish a version of Calder´on’s reproducing formula, namely, f = X j,k ∈ Z X R ∈ R ( N,j,k ) | R | Z − j +1 − N − j − N Z − k +1 − N − k − N f ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s (cent( R )) dss drr ˜ ϕ j,k ( · , cent( R )) , where the equality holds in L ( H ν ) and in the flag test function space M , as defined in [32, Definition3], R ( N, j, k ) is the collection of all adapted rectangles R for which width( R ) = 2 − j − N and height( R ) =2 − max { j,k }− N , N being a large positive integer which will be determined later, and ˜ ϕ j,k ( · , cent( R )) hasthe same size, regularity and cancellation properties as ϕ (1)2 − j ∗ (2) ϕ (2)2 − k ( g ). The proof is now standard,and follows the main lines of the proof of [32, Theorem 3] (see also [28, Theorem 2.5]), so we onlysketch it. We now take f in the test function space M .First, for an appropriate choice of ˜ ϕ (1) r and ˜ ϕ (2) s , as discussed in Section 2.3, we discretize f as f ( g ) = Z Z R + × R + f ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s ∗ (1) ˜ ϕ (1) r ∗ (2) ˜ ϕ (2) s ( g ) dss drr = X j,k ∈ Z Z − j − N − j − N Z − k − N − k − N X R ∈ R ( N,j,k ) Z R f ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s ( g ′ ) ˜ ϕ (1) r ∗ (2) ˜ ϕ (2) s (( g ′ ) − g ) dg ′ dss drr =: T ( f )( g ) + R ( f )( g )for all g ∈ H ν and all f ∈ L ( H ν ), where T ( f )( g ) is defined to be T ( f )( g ) := X j,k ∈ Z X R ∈ R ( N,j,k ) | R | Z − j − N − j − N Z − k − N − k − N f ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s (cent( R )) dss drr | R | Z R ˜ ϕ (1)2 − j ∗ (2) ˜ ϕ (2)2 − k (( g ′ ) − g ) dg ′ and R ( f )( g ) is the remainder. In particular, T = I − R , where I denotes the identity operator.Second, by verifying the size and regularity conditions we deduce that R maps the test functionspace M into itself with the norm kR ( f ) k M ≤ C − N k f k M . Then, by choosing N large enough, we may ensure that C − N <
1, which implies that T is invertibleon M and kT − k M = k ( I − R ) − k M ≤ (cid:13)(cid:13)(cid:13) ∞ X i =0 R i (cid:13)(cid:13)(cid:13) M ≤ ∞ X i =0 kRk i M < ∞ (here k·k M denotes the operator norm of an operator on M ). Hence f ( g ) = T − T ( f )( g )= X j,k ∈ Z X R ∈ R ( N,j,k ) | R | Z − j +1 − N − j − N Z − k +1 − N − k − N f ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s (cent( R )) dss drr ˜ ϕ j,k ( g, cent( R )) , (2.27)where ˜ ϕ j,k ( g, cent( R )) := T − (cid:18) | R | Z R ϕ (1)2 − j ∗ (2) ϕ (2)2 − k (( g ′ ) − · g ) dg ′ (cid:19) . By duality, we also see that this reproducing formula holds in the sense of distribution (cid:0) M (cid:1) ′ .27y using the reproducing formula (2.27), and an almost orthogonality estimate, we can deducethe flag Plancherel–P´olya inequality (the analogue of [28, Theorem 2.13] in the Euclidean setting) (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) X j,k ∈ Z X R ∈ R ( N,j,k ) Z − j − N − j − N Z − k − N − k − N sup g R ∈ CR (cid:12)(cid:12)(cid:12) f ∗ (1) ψ (1) r ∗ (2) ψ (2) s ( g R ) (cid:12)(cid:12)(cid:12) dss drr χ R ( · ) (cid:19) / (cid:13)(cid:13)(cid:13)(cid:13) L ( H ν ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) X j,k ∈ Z X R ∈ R ( N,j,k ) Z − j − N − j − N Z − k − N − k − N inf g R ∈ R (cid:12)(cid:12)(cid:12) f ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s ( g R ) (cid:12)(cid:12)(cid:12) dss drr χ R ( · ) (cid:19) / (cid:13)(cid:13)(cid:13)(cid:13) L ( H ν ) . Then by using the flag Plancherel–P´olya inequality just proved, we see that for f ∈ L ( H ν ), (cid:13)(cid:13)(cid:13)(cid:13) S F, cts , ϕ ( f )( · ) (cid:13)(cid:13)(cid:13)(cid:13) L ( H ν ) = Z H ν (cid:18)Z Z R + × R + (cid:12)(cid:12) f ∗ (1) ψ (1) r ∗ (2) ψ (2) s ( g ) (cid:12)(cid:12) drr dss (cid:19) / dg = Z H ν (cid:26) X j,k ∈ Z X R ∈ R ( N,j,k ) Z − j +1 − N − j − N Z − k +1 − N − k − N | f ∗ (1) ψ (1) r ∗ (2) ψ (2) s ( g ) | dss drr χ R ( g ) (cid:27) / dg ≤ Z H ν (cid:26) X j,k ∈ Z X R ∈ R ( N,j,k ) Z − j +1 − N − j − N Z − k +1 − N − k − N sup g R ∈ R | f ∗ (1) ψ (1) r ∗ (2) ψ (2) s ( g R ) | dss drr χ R ( g ) (cid:27) / dg ≈ Z H ν (cid:26) X j,k ∈ Z X R ∈ R ( N,j,k ) Z − j +1 − N − j − N Z − k +1 − N − k − N inf g R ∈ R | f ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s ( g R ) | dss drr χ R ( g ) (cid:27) / dg ≤ Z H ν (cid:18)Z Z R + × R + (cid:12)(cid:12) f ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s ( g ) (cid:12)(cid:12) drr dss (cid:19) / dg, and the proof of Lemma 2.25 is complete. In this section we focus on the Hardy space defined via atomic decompositions. We first examine thedefinition and properties of the space in more detail than may be found in most previous studies,and then show that if f ∈ H F, atom ( H ν ), then S F, area , ϕ ( f ) ∈ L ( H ν ), M F, gmax ( f ) ∈ L ( H ν ), and K ( f ) ∈ L ( H ν ) when K is a simple singular integral operator. We recall the definition of the atomic Hardy space, and make it more precise.
Definition 3.1.
Fix integers
M > ν/ N ≥ κ ∈ (1 , ∞ ). An atom is afunction a ∈ L ( H ν ) such that there exists an open subset Ω of H ν of finite measure | Ω | and functions a R in L ( H ν ), called particles , and b R in Dom(∆ (1) M ∆ (2) N ) in L ( H ν ) for all R ∈ M (Ω) such that(A1) a R = ∆ (1) M ∆ (2) N b R and supp b R ⊆ R ∗ = T ( g, κq, κ h ), where g = cent( R ), q = width( R ) and h = height( R );(A2) for all sign sequences σ : M (Ω) → {± } , the sum P R ∈ M (Ω) σ R a R converges in L ( H ν ), to a σ say, and k a σ k L ( H ν ) ≤ | Ω | − / ; (3.1)(A3) a = P R ∈ M (Ω) a R . 28e say that f ∈ L ( H ν ) has an atomic decomposition if we may write f as a sum P j ∈ N λ j a j ,converging in L ( H ν ), where P j ∈ N | λ j | < ∞ and each a j is an atom; we write f ∼ P j ∈ N λ j a j toindicate that P j ∈ N λ j a j is an atomic decomposition of f . The space H F, atom ( H ν ) is defined to be thelinear space of all f ∈ L ( H ν ) that have atomic decompositions, with norm k f k H F, atom ( H ν ) := inf (cid:26) X j ∈ N | λ j | : f ∼ X j ∈ N λ j a j (cid:27) . (3.2)A few comments are in order. First, by a randomisation argument, the norm condition on sums(A2) implies the condition P R ∈ M (Ω) k a R k L ( H ν ) ≤ | Ω | − that is standard in the literature, and theusual condition that k a k L ( H ν ) ≤ | Ω | − / . Our condition is just as easy to verify, and is more useful; itis a quantitative form of unconditional convergence. In particular, it shows that the sum converges inany order, and with any regroupings, and implies a similar inequality to (3.1) when σ is just a boundedsequence. See, for instance, [41, Section 1.c] for information about unconditional convergence.Third, the enlargement R ∗ of R has the same centre as R , but is bigger and smoother than R . Ifwe need to be more precise, we say that an atom a and the Hardy space as above are a (1 , , M, N, κ )-atom and the (1 , , M, N, κ )-atomic Hardy space. We will discuss the dependence of the atomic Hardyspace on κ in the following section.We note that if a ∈ L ( H ν ), and a = ∆ M (1) ∆ M (2) b , where b ∈ L ( H ν ) and supp b ⊆ R for some R ∈ R ,then a = k a k L ( H ν ) | R | / c , where c = k a k − L ( H ν ) | R | − / a , and c is an atom, so that k a k H F, atom ( H ν ) ≤ | R | / k a k L ( H ν ) . (3.3) In this section, we clarify some properties of the atomic Hardy space in the flag setting. We examinethe dependence on the enlargement parameter, we show that there is a dense subspace with atomicdecompositions that converge in L ( H ν ) as well as in L ( H ν ), and we show that a certain linear ornonnegative-real-valued sublinear operators that are bounded on L ( H ν ) and which send particles into L ( H ν ) and satisfy certain decay estimates are bounded from H F, atom ( H ν ) into L ( H ν ).In the classical theory, Hardy spaces may be defined using atoms which are naturally L p functions,where p may be any index greater than 1, including ∞ , and the convergence considered was a simple ℓ sum. In later versions of the theory, matters became more complicated, and in particular, in someHardy spaces associated to rough differential operators, it is not at all clear whether the norm is givenby the expression (3.2). One aim of this section is to establish that we do have such a representation.The first point is that the L ( H ν ) norm conditions on the particles a R imply norm conditions onthe functions b R . Corollary 3.2. If R ∈ R , a R ∈ L ( H ν ) , b R ∈ Dom(∆ (1) M ∆ (2) N ) in L ( H ν ) , a R = ∆ (1) M ∆ (2) N b R and supp b R ⊆ R ∗ = T ( g, κq/ , κ (4 h + q ) / , where g = cent( R ) , q = width( R ) and h = height( R ) ,as in Definition 3.1, then k b R k L ( H ν ) . κ q M h N k a R k L ( H ν ) . Proof.
Iterate the first inequality of the previous lemma N times, and the second M times.In this vein, we should point out that subelliptic regularity (see, for instance, [20, Theorem 6.1])implies that, for any atom a , there exists a function b in L ( H ν ) such that a = ∆ M (1) ∆ N (2) b . If Ω is29ounded or if N = 0, then it is immediately apparent that b ∈ L ( H ν ), but if Ω is unbounded and N = 0, then this is not so.Our second remark is that in the definition of an atom, we may parametrise particles by general,not necessarily maximal, adapted rectangles, and we may group together the particles in many ways. Lemma 3.3.
Fix integers M ≥ and N ≥ and a real number κ > . Let Ω be an open subset of H ν of finite measure and R (Ω) be the set of adapted subrectangles of Ω . Suppose that there exist functions a R in L ( H ν ) and b R in Dom(∆ (1) M ∆ (2) N ) in L ( H ν ) for all R ∈ R (Ω) such that(A1) a R = ∆ (1) M ∆ (2) N b R and supp b R ⊆ R ∗ ;(A2) for all sign sequences σ : M (Ω) → {± } , the sum P R ∈ M (Ω) σ R a R converges in L ( H ν ) , to a σ say, and k a σ k L ( H ν ) ≤ | Ω | − / ; (A3) a = P R ∈ M (Ω) a R .Suppose that S is a subcollection of R (Ω) and † : R (Ω) → S is a mapping such that R ⊆ R † . Thenfor each S ∈ S , the sum P R ∈ R (Ω): R † = S a R converges in L ( H ν ) , to ˜ a S , say, and P R ∈ R (Ω): R † = S b R converges in L ( H ν ) , to ˜ b S , say. Further,(B1) ˜ a S = ∆ (1) M ∆ (2) N ˜ b S and supp b S ⊆ S ∗ ;(B2) for all sign sequences σ : S → {± } , the sum P S ∈ S σ S ˜ a S converges in L ( H ν ) , to ˜ a σ say, and k ˜ a σ k L ( H ν ) ≤ | Ω | − / ; (B3) a = P S ∈ S ˜ a S .In particular, any function a for which (A1) to (A3) hold is an atom.Proof. By unconditional convergence, for each S ∈ S , the sum P R ∈ R (Ω): R † = S a R converges in L ( H ν ),to ˜ a S , say, and by Lemma 2.23, P R ∈ R (Ω): R † = S b R converges in L ( H ν ), to ˜ b S , say, and ∆ M (1) ∆ N (2) ˜ b S =˜ a S . Now R ⊆ R † , so R ∗ ⊆ ( R † ) ∗ , and supp(˜ b S ) ⊆ S ∗ . Moreover, any sign function σ : S →{± } determines a sign function ˜ σ : R (Ω) → {± } by the rule ˜ σ R = σ R † . Again by unconditionalconvergence, P S ∈ S σ S ˜ a S converges to P R ∈ M (Ω) ˜ σ R a R in L ( H ν ) and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X S ∈ S σ S ˜ a S (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( H ν ) ≤ | Ω | − / for all sign sequences σ : S → {± } , and a = P S ∈ M (Ω) S ˜ a S in L ( H ν ).In particular, for each adapted rectangle R ∈ R (Ω), there is a unique widest adapted rectangle S of maximal volume in M (Ω) such that R ⊆ S . Indeed, if R ⊆ S ⊆ Ω and R ⊆ S ⊆ Ω, where S and S are adapted rectangles and width( S ) = width( S ), then R ⊆ S ∪ S ⊆ Ω, and S ∪ S is alsoa adapted rectangle. We denote this widest maximal adapted rectangle S by R ↑ . As the mapping R R ↑ has the property R ⊆ R ↑ , we may take S to be M (Ω), and deduce that a is an atom.Our third remark is that the set Ω involved in the definition of an atom need not be open.30 orollary 3.4. Fix integers M ≥ and N ≥ and a real number κ > . Let S be a countable setof adapted subrectangles, let Ω = S R ∈ S S , and suppose that Ω has finite measure. Suppose that thereexist functions a R in L ( H ν ) and b R in Dom(∆ (1) M ∆ (2) N ) in L ( H ν ) for all R ∈ S such that(A1) a R = ∆ (1) M ∆ (2) N b R and supp b R ⊆ R ∗ ;(A2) for all sign sequences σ : M (Ω) → {± } , the sum P R ∈ M (Ω) σ R a R converges in L ( H ν ) , to a σ say, and k a σ k L ( H ν ) ≤ | Ω | − / ; (A3) a = P R ∈ M (Ω) a R .Then (1 + ε ) − a is an atom for all ε ∈ R + .Proof. Since | Ω | is finite, for all positive ε , there is an open subset Ω ε of H ν such that | Ω ε | < (1 + ε ) | Ω | .The rectangles R ∈ S all lie in R (Ω ε ), and the previous lemma implies that (1 + ε ) − a is an atom.Henceforth, we replace the requirement that Ω be an open set by the assumption that it is acountable union of adapted rectangles.Fourth, we do not need to consider adapted rectangles. Corollary 3.5.
Fix integers M ≥ and N ≥ . Suppose that there exist functions a k in L ( H ν ) and b k in Dom(∆ (1) M ∆ (2) N ) in L ( H ν ) and tubes T k for all k ∈ N such that(A1) a k = ∆ (1) M ∆ (2) N b k and supp b k ⊆ T k ;(A2) for all sign sequences σ : N → {± } , the sum P k ∈ N σ k a k converges in L ( H ν ) , to a σ say, and k a σ k L ( H ν ) ≤ (cid:12)(cid:12)(cid:12)(cid:12) [ k ∈ N T k (cid:12)(cid:12)(cid:12)(cid:12) − / ( in particular, S k ∈ N T k has finite measure ) ;(A3) a = P k ∈ N a k .Then a is a (1 , , M, N, (2 ν + 1) ) atom.Proof. By Lemma 2.6, for each k ∈ N , we can find an adapted rectangle R k such that R k ⊆ T k ⊆ R ∗ k ,where the enlargement parameter κ is taken to be (2 ν + 1) . We take Ω to be S k ∈ N R k . After groupingtogether all a k that give rise to the same R k , we are in a situation that satisfies the hypotheses ofLemma 3.3, and so a is an atom.Next, atomic Hardy space does not depend on the enlargement parameter κ (though the normmay well do so). Lemma 3.6.
Suppose that κ > κ > and κ − ≥ κ/ (2 ν + 1) . Then every (1 , , M, N, κ ) -atom is theproduct of a geometric constant and a (1 , , M, N, κ ) -atom. Consequently, the (1 , , M, N, κ ) -atomicHardy space and the (1 , , M, N, κ ) -atomic Hardy space coincide.Proof. In this proof, we include the enlargement parameter κ in the notation, and write R ∗ ,κ insteadof R ∗ for g · B (1) ( o, κq/ · B (2) (0 , κ ( q + 4 h ) / g = cent( R ), q = width( R ), and h = height( R ).Suppose that a is a (1 , , M, N, κ )-atom associated to the open set Ω. Then we may find functions a R and b R in L ( H ν ) for all R ∈ M (Ω) such that 31A1) a R = ∆ (1) M ∆ (2) N b R and supp b R ⊆ R ∗ ,κ ;(A2) for all sign sequences σ : M (Ω) → {± } , the sum P R ∈ M (Ω) σ R a R converges in L ( H ν ), to a σ say, and k a σ k L ( H ν ) ≤ | Ω | − / ;(A3) a = P R ∈ M (Ω) a R .For R ∈ M (Ω), write R † for the unique adapted rectangle that contains R and is a translate of δ ν +1 ( R ), and let Ω † be the set S R ∈ M (Ω) R † . Then, as shown in Corollary 2.12, (cid:12)(cid:12) Ω † (cid:12)(cid:12) . | Ω | . Further,for such R and R † , write g = cent( R ), g † = cent( R † ), q = width( R ) and h = height( R ); then g is inthe interior of R † , and (2.13) implies that R ∗ ,κ = g · B ( o, κq/ · B (2) (0 , κ ( q + 4 h ) / ⊆ g † · B ( o, (2 ν + 1) q/ · B (2) (0 , (2 ν + 1) ( q + 4 h ) / · B ( o, κq/ · B (2) (0 , κ ( q + 4 h ) / ⊆ g † · B ( o, (2 ν + 1 + κ ) q/ · B (2) (0 , ((2 ν + 1) + κ )( q + 4 h ) / ⊆ g † · B ( o, κ (2 ν + 1) q/ · B (2) (0 , κ (2 ν + 1) ( q + 4 h ) / R † ) ∗ ,κ , since κ − ≥ κ/ (2 ν + 1).We take S to be the collection { R † : R ∈ M (Ω) } , and for S ∈ S , we write˜ a S = X R ∈ M (Ω): R † = S a R and ˜ b S = X R ∈ M (Ω): R † = S b R , as in the previous lemma. Clearly, supp b R ⊆ ( R † ) ∗ ,κ , whence supp ˜ b S ⊆ S ∗ ,κ , By Lemma 3.3,( | Ω | / | ˜Ω | ) / a is a (1 , , M, N, κ )-atom.It follows that the (1 , , M, N, κ )-atomic Hardy space is a subspace of the (1 , , M, N, κ )-atomicHardy space; since the converse is trivial, these spaces coincide. Corollary 3.7.
As a topological vector space, the (1 , , M, N, κ ) -atomic Hardy space is independentof the parameter κ in (1 , ∞ ) . In light of this corollary, we shall usually abbreviate (1 , , M, N, κ )-atom to (1 , , M, N )-atom.Our sixth remark is that we used the family of adapted rectangles R in defining atoms. Since adilate or a left translate of an adapted rectangle need not be an adapted rectangle, it is possible that H F, atom ( H ν ) might not be dilation or (left) translation invariant. Since the other Hardy spaces definedin Section 1 are evidently dilation and left translation invariant, this would mean that these spacescould not coincide with the atomic space. Fortunately, Corollary 3.5 implies that this is not the case.We recall two definitions: for a function f on H ν , g ∈ H ν , and r ∈ R + , the left translate g f of f by g and the normalised dilate f r of f by r are given by g f ( g ′ ) = f ( g − g ′ ) and f r ( g ′ ) = r − D f ( δ /r g ′ ) ∀ g ′ ∈ H ν . Corollary 3.8.
Suppose that f ∈ H F, atom ( H ν ) . Then the following hold:1. for all g ∈ H ν , the translate g f is in H F, atom ( H ν ) and k g a k H F, atom ( H ν ) ≤ (2 ν + 1) D/ ’2. as g → o in H ν , g f → f in H F, atom ( H ν ) ; . for all r ∈ R + , the normalised dilate f r is in H F, atom ( H ν ) and k f r k H F, atom ( H ν ) ≤ (2 ν + 1) D/ ;4. as r → in R + , f r → f in H F, atom ( H ν ) .Proof. The atoms of Corollary 3.5 are based on tubes, and translates and dilates of tubes are tubes.To see that translations and dilations act continuously, we note that for a single particle supportedin a tube, and all g close to the identity of H ν , g a R − a R is supported in a slightly larger tube, and k g a R − a R k L ( H ν ) → g → o . We leave the reader the task of putting together the estimatesfor particles to obtain estimates for atoms and putting together the estimates for atoms to obtainestimates for general elements of H F, atom ( H ν ). Lemma 3.9.
Suppose that M > ν and N ≥ , that ϕ is a smooth nonnegative-real-valued radial function with support in B (1) ( o, such that R H ν ϕ ( g ) dg = 1 , and that ϕ r ( · ) = r − D ϕ ( δ /r · ) . Then a ∗ (1) ϕ r → a as r → in L ( H ν ) as well as in H F, atom ( H ν ) . Further, there is a constant C ϕ withthe properties that, for all ε ∈ R + , each function a ∗ (1) ϕ r has an atomic decomposition P k c k,r a k,r such that1. P k c k,r a k,r converges in L ( H ν ) as well as in L ( H ν ) ;2. P k | c k,r | ≤ C ϕ for all r ∈ R + ;3. lim r → a ∗ (1) ϕ r = a in L ( H ν ) as well as in H F, atom ( H ν ) , and lim r → P k | c k,r | ≤ ε .Proof. We prove items 1 and 2 first, and then consider item 3.Suppose that a ∈ L ( H ν ), that Ω is an open subset of H ν , and that there exist functions a R in L ( H ν ) and b R in Dom(∆ (1) M ∆ (2) N ) in L ( H ν ) for all R ∈ M (Ω), such that(A1) a R = ∆ (1) M ∆ (2) N b R and supp b R ⊆ R ∗ ;(A2) for all σ : M (Ω) → {± } , the sum P R ∈ M (Ω) σ R a R converges in L ( H ν ), to a σ say, and k a σ k L ( H ν ) ≤ | Ω | − / ;(A3) a = P R ∈ M (Ω) a R .Suppose that R ∈ M (Ω) and that a R and b R are as above. We denote g := cent( R ), q := width( R )and h := height( R ); then R ∗ = T ( g, κq, κ h ); further, supp( b R ∗ (1) ϕ r ) ⊆ T ( g, κq + r, κ h ), and a R ∗ (1) ϕ r = (∆ M (1) ∆ N (2) b R ) ∗ (1) ϕ r = (∆ N (2) b R ) ∗ (1) ∆ M (1) ϕ r = b R ∗ (1) (∆ M (1) ∆ N (2) ϕ r ) , since ∆ (1) and ∆ (2) correspond to right convolutions with radial distributions, and convolution iscommutative for radial functions and distributions. We write R ∈ M if r ≤ q , R ∈ M if q ≤ r ≤ ( q + h ) / and R ∈ M if ( q + g ) / ≤ r , and we define a = P R ∈ M a R , a = P R ∈ M a R and a = P R ∈ M a R . Then a , a and a are all atoms associated to Ω, which we are going to considerseparately. In light of (A2), and the fact that the decomposition of M (Ω) into M , M and M depends on r , it is apparent that there are functions η ( r ), η ( r ) and η ( r ), taking values in [0 ,
1] suchthat (cid:13)(cid:13)(cid:13) X R ∈ M k σ R a R (cid:13)(cid:13)(cid:13) L ( H ν ) ≤ η k ( r ) | Ω | − / σ : M k → {± } , as k varies from 1 to 3. It is also clear from the unconditionalconvergence of the series in (A2) that η ( r ) → η ( r ) → η ( r ) → r → to be S R ∈ M R † , where R † is the unique adapted rectangle containing R that is atranslate of δ ν +1 R ; then | Ω | . | Ω | by Corollary 2.12. For each R ∈ M , let R ↑ be the unique widestrectangle S ∈ M (Ω ) that contains R . Evidently a ∗ (1) ϕ r = X R ∈ M a R ∗ (1) ϕ R = X S ∈ M (Ω ) ˜ a S , where ˜ a S = X R ∈ M (Ω): R ↑ = S a R ∗ (1) ϕ R , and all these sums converge unconditionally, as in the proof of Lemma 3.3. Similarly, P R : R ↑ = S b R ∗ (1) ϕ r converges unconditionally, to ˜ b S , say, and ˜ a S = ∆ M (1) ∆ N (2) ˜ b S and supp ˜ b S ⊆ S ∗ . Further, for any signfunction σ : M (Ω ) → {± } we may define a sign function ˜ σ : M → {± } so that ˜ σ R = σ R ↑ , andthen X S ∈ M (Ω ) σ S X R : R ↑ = S a R ∗ (1) ϕ r ! = X R ∈ M ˜ σ R a R ∗ (1) ϕ r , whence (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X S ∈ M (Ω ) σ S ˜ a S (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( H ν ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X R ∈ M ˜ σ R a R (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( H ν ) ≤ η ( r ) | Ω | − / . It follows that a is ( | Ω | / | Ω | ) / multiplied by an atom associated to Ω .We deal with a by grouping the rectangles in M according to their widths, that is, q . Bydefinition, if R ∈ M , then q ≤ r ≤ ( q + h ) / . Further, since a R ∗ (1) ϕ r = ∆ N (2) b R ∗ (1) ∆ M (1) ϕ r , we have (cid:13)(cid:13) a R ∗ (1) ϕ r (cid:13)(cid:13) L ( H ν ) ≤ (cid:13)(cid:13)(cid:13) ∆ N (2) b R (cid:13)(cid:13)(cid:13) L ( H ν ) (cid:13)(cid:13)(cid:13) ∆ M (1) ϕ r (cid:13)(cid:13)(cid:13) L ( H ν ) . ϕ,N q M r M k a R k L ( H ν ) , (3.4)by Lemma 2.23, and supp( b R ∗ (1) ϕ r ) ⊆ R ∗ · B (1) ( o, r ) ⊆ g · B (1) (2 κr ) · B (2) ( κ h ). For j ∈ Z such that(2 ν + 1) j ≤ r , we write M ,j for the family of all R ∈ M for which q = (2 ν + 1) j . The rectanglesin M ,j are pairwise disjoint because they are maximal. This disjointness implies that the number ofoverlapping supports R ∗ · B (1) ( o, r ) where R ∈ M ,j that contain a given point g ∈ H ν , which is thenumber of these rectangles R that meet B (1) ( g, r ), is bounded by a constant multiple of ( r/q ) ν , andso for any sign sequence σ : M ,j → {± } , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X R ∈ M ,j σ R a R ∗ (1) ϕ r (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( H ν ) . (cid:18) rq (cid:19) ν X R ∈ M ,j (cid:13)(cid:13) a R ∗ (1) ϕ r (cid:13)(cid:13) L ( H ν ) / . ϕ,N (cid:16) qr (cid:17) M − ν X R ∈ M ,j k a R k L ( H ν ) / h (cid:16) qr (cid:17) M − ν (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X R ∈ M ,j ˜ σ R a R (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( H ν ) ≤ (cid:16) qr (cid:17) M − ν η ( r ) | Ω | − / , (3.5)34y (3.4), the almost disjointness of the supports of the a R in consideration, and the fact that a is anatom.Finally, for R ∈ M ,j , we choose R + to be the smallest adapted rectangle such that R ⊆ R + and R ∗ · B (1) ( o, r ) ⊆ ( R + ) ∗ . For this, it is sufficient that width( R + ) ≥ r/ ( κ − | R + | . r ν h . ( r/q ) ν | R | . Define Ω ,j = S R ∈ M ,j R + ; then | Ω ,j | ≤ X R ∈ M ,j (cid:12)(cid:12) R + (cid:12)(cid:12) . (cid:18) rq (cid:19) ν X R ∈ M ,j | R | ≤ (cid:18) rq (cid:19) ν | Ω | . (3.6)We write a ,j = P S ∈ R (Ω ,j ) ˜ a S , where ˜ a S = P R ∈ M ,j : R + = S a R . By Lemma 3.3, (3.5) and (3.6), for anysign sequence σ : R (Ω ,j ) → {± } , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X S ∈ R (Ω ,j ) σ S ˜ a S (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( H ν ) . ϕ,N (cid:16) qr (cid:17) M − ν η ( r ) | Ω ,j | − / , and so a ,j is a multiple of ( q/r ) M − ν η ( r ) times an atom associated with Ω ,j ; since q = (2 ν + 1) j and P j ((2 ν + 1) j /r ) M − ν is bounded when we sum over those integers j for which (2 ν + 1) j ≤ r , wehave controlled a .In the third case, where ( q + h ) / ≤ r , we observe that supp( b ∗ (1) ϕ r ) ⊆ g · B (1) (3 r ), and (cid:13)(cid:13) a ∗ (1) ϕ r (cid:13)(cid:13) L ( H ν ) = (cid:13)(cid:13)(cid:13) ∆ M (1) ∆ N (2) b ∗ (1) ϕ r (cid:13)(cid:13)(cid:13) L ( H ν ) = (cid:13)(cid:13)(cid:13) b ∗ (1) ∆ M (1) ∆ N (2) ϕ r (cid:13)(cid:13)(cid:13) L ( H ν ) ≤ k b k L ( H ν ) (cid:13)(cid:13)(cid:13) ∆ M (1) ∆ N (2) ϕ r (cid:13)(cid:13)(cid:13) L ( H ν ) . q M ( q + h ) N k a k L ( H ν ) r − M − N = q M ( q + h ) N r M +4 N k a k L ( H ν ) , (3.7)again by Lemma 2.23. A very similar argument to that used for a deals with a ; a double sum isneeded to manage the widths and the heights of the rectangles and (3.7) replaces (3.4).Finally, it is evident that a ∗ (1) ϕ r → a in L ( H ν ) as r → a ∗ (1) ϕ r → a in H F, atom ( H ν ) as r → X R ∈ M ( a R − a R ∗ (1) ϕ r ) + X R ∈ M a R − X R ∈ M a R ∗ (1) ϕ r + X R ∈ M a R − X R ∈ M a R ∗ (1) ϕ r . The second and the fourth sum tend to 0 as r →
0+ because of the unconditional convergence in(A2); the third and fifth sums tend to 0 as r →
0+ because η ( r ) → η ( r ) → r → r →
0+ by considering the supports of b R and b R ∗ (1) ϕ r to be the rectangle R + and observing that (cid:13)(cid:13) a R − a R ∗ (1) ϕ r (cid:13)(cid:13) L ( H ν ) → r → Proposition 3.10.
Suppose that A is a linear operator, or a nonnegative sublinear operator, thatsatisfies a strong type (2 , bound kA f k L ( H ν ) ≤ C k f k L ( H ν ) or a weak-type (2 , bound |{ g ∈ H ν : |A f ( g ) | > λ }| ≤ C λ k f k L ( H ν ) ∀ λ ∈ R + or all f ∈ L ( H ν ) . Let M, N ∈ Z + .(a) Suppose also that there exist C, δ , δ ∈ R + such that Z ( S ∗ ) c |A a R ( g ) | dg ≤ Cρ δ ,δ ( R, S ) | R | / k a R k L ( H ν ) (3.8) for all S ∈ R such that R ⊆ S ∈ R and all particles a R ∈ L ( H ν ) such that a R = ∆ M (1) ∆ N (2) b R where b R ∈ L ( H ν ) and supp b R ⊆ R ∗ and ρ δ ,δ ( R, S ) := (cid:18) height( R )height( S ) (cid:19) δ + (cid:18) width( R )width( S ) (cid:19) δ . (3.9) Then there is a constant C such that kA a k L ( H ν ) ≤ C (3.10) for all (1 , , M, N ) -atoms a .(b) Suppose also that (3.10) holds. Then A maps H F, atom ( H ν ) ∩ L ( H ν ) into L ( H ν ) , and kA f k L ( H ν ) ≤ C k f k H F, atom ( H ν ) ∀ f ∈ H F, atom ( H ν ) ∩ L ( H ν ) . (3.11) Hence A may be extended uniquely by continuity to a bounded operator from H F, atom ( H ν ) to L ( H ν ) that satisfies the same inequality for all f ∈ H F, atom ( H ν ) .Proof. Take a (1 , , M, N, κ )-atom a associated to an open set Ω of finite measure, and write a ∼ P R ∈ M (Ω) a R as in Definition 3.1. Define the set Υ byΥ = { g ∈ H ν : M s Ω ≥ / D } . If R is any maximal adapted subrectangle of Ω, and g ∈ S ∗ = 100 R , then 1 / D ≤ M s R ( g ) ≤M s Ω ( g ), so that S ∗ ⊆ Υ. As the flag maximal function is L bounded, Chebyshev’s inequality showsthat | Υ | . k Ω k L ( H ν ) D . | Ω | , where the implicit constants are independent of Ω. Now kA ( a ) k L ( H ν ) = Z Υ |A ( a )( g ) | dg + Z Υ c |A ( a )( g ) | dg. From the estimate k a k L ( H ν ) ≤ | Ω | − / , the L boundedness of A , and H¨older’s inequality Z Υ |A ( a )( g ) | dg ≤ | Υ | / kA ( a ) k L ( H ν ) . | Ω | / k a k L ( H ν ) ≤ , where the implicit constant is geometric. Hence, to prove (3.10), it suffices to prove that there exists aconstant C such that R Υ c |A ( a )( g ) | dg ≤ C . We write a as P R ∈ M (Ω) a R , and then A a = P R ∈ M (Ω) A a R ,where the sum converges in L ( H ν ) by L -boundedness, so Z Υ c |A ( a )( g ) | dg ≤ X R ∈ M (Ω) Z Υ c |A ( a R )( g ) | dg ≤ X R ∈ M (Ω) Z ( R †∗ ) c |A ( a R )( g ) | dg.
36y (3.8), the Cauchy–Schwarz inequality and Journ´e’s lemma (Lemma 2.13), it follows that kA ( a ) k L ( H ν ) . X R ∈ M (Ω) ρ δ ,δ ( R, S ) | R | / X R ∈ M (Ω) k a R k L ( H ν ) / . | Ω | / | Ω | − / = 1 , as required.To prove (b), it will suffice to prove (3.11). Take f ∈ H F, atom ( H ν ) ∩ L ( H ν ) and ε ∈ R + . Thenwe may write f ∼ P j λ j a j , where the a j are atoms and P j | λ j | < k f k H F, atom ( H ν ) + ε . Take a smoothnonnegative-real-valued radial function ϕ with support in B (1) ( o,
1) such that R H ν ϕ r ( g ) dg = 1, andset ϕ r ( · ) = r − D ϕ ( δ /r · ), as in Lemma 3.9.Now f = lim r → f ∗ (1) ϕ r in L ( H ν ) and so A f = lim r → A ( f ∗ (1) ϕ r ) in L ( H ν ). Further, thesum P j λ j a j converges in L ( H ν ), and so the sum P j λ j a j ∗ (1) ϕ r converges in L ( H ν ), and |A f | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) lim r → lim J →∞ A J X j =1 λ j a j ∗ (1) ϕ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ lim r → lim J →∞ J X j =1 (cid:12)(cid:12) λ j A ( a j ∗ (1) ϕ r ) (cid:12)(cid:12) = lim r → ∞ X j =1 | λ j | (cid:12)(cid:12) A ( a j ∗ (1) ϕ r ) (cid:12)(cid:12) . By Lemma 3.9, a j ∗ (1) ϕ r ∼ P k c j,k,r a j,k,r where the sum converges in L ( H ν ); further, P k | c j,k,r | ≤ C for all r ∈ R + ; and lim r → P j,k ∈ Z | c j,k,r | = 1. Then (cid:12)(cid:12) A ( a j ∗ (1) ϕ r ) (cid:12)(cid:12) ≤ P k | c j,k,r | |A a j,k,r | , so |A f | ≤ lim r → ∞ X j =1 | λ j | X k | c j,k,r | |A a j,k,r | ≤ lim r → X j,k ∈ Z | λ j | | c j,k,r | |A a j,k,r | whence kA f k L ( H ν ) ≤ lim r → C X j,k ∈ Z | λ j | | c j,k,r | ≤ C ( k f k H F, atom ( H ν ) + ε )by dominated convergence. Since ε is arbitrary, (3.11) holds. H F, atom ( H ν ) ⊆ H F, area ( H ν ) In this section, we consider the Lusin–Littlewood–Paley operator S F, area , ϕ , and show that it is boundedon the atomic Hardy space. We first recall the definition of this operator. We suppose that the integers M and N satisfy M > D/ N ≥
1, and consider a pair ˜ ϕ of Poisson-bounded functions, as inDefinition 2.20. We define ϕ (1) = ∆ (1) ˜ ϕ (1) and ϕ (2) = ∆ (2) ˜ ϕ (2) , and take ϕ (1) r to be r − D ϕ (1) ( δ /r · ) and ϕ (2) s to be s − ϕ (2) ( · /s ). Further, we define χ (1) r to be the L ( H ν )-normalised characteristic function ofthe ball B (1) ( o, r ) and χ (2) s to be the L ( R )-normalised characteristic function of the ball B (2) (0 , s ). Definition.
Suppose that ϕ (1) , ϕ (2) , χ (1) and χ (2) are as above. For f ∈ L ( H ν ), we define theLusin–Littlewood–Paley area function S F, area , ϕ ( f ) associated to ϕ (1) and ϕ (2) by S F, area , ϕ ( f )( g ) := (cid:18)Z Z R + × R + (cid:12)(cid:12) f ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s (cid:12)(cid:12) ∗ (1) χ (1) r ∗ (2) χ (2) s ( g ) drr dss (cid:19) / g ∈ H ν , and we define the Hardy space H F, area , ϕ ( H ν ), usually abbreviated to H F, area ( H ν ), tobe the set of all f ∈ L ( H ν ) for which kS F, area , ϕ ( f ) k L ( H ν ) < ∞ , with norm k f k H F, area , ϕ ( H ν ) := kS F, area , ϕ ( f ) k L ( H ν ) . Theorem 3.11.
Suppose that
M > D/ , N ≥ and κ > , and that the sublinear operator S F, area , ϕ and the space H F, area , ϕ ( H ν ) are as defined above. Then H F, atom ,M,N,κ ( H ν ) ⊆ H F, area , ϕ , and k f k H F, area ( H ν ) . M,N, ϕ k f k H F, atom ( H ν ) ∀ f ∈ H F, atom ( H ν ) . Proof.
The proof involves two steps: first we suppose that ϕ (1) and ϕ (2) are supported in the unitballs in H ν and in R , and show that there exists δ , δ ∈ R + such that, for all particles a R for which b R is supported in R ∗ , Z ( S ∗ ) c |S F, area , ϕ ( a R )( g ) | dg . δ ,δ , ϕ ρ δ ,δ ( R, S ) | R | / k a R k L ( H ν ) , where S is any adapted rectangle that contains R ; this step is carried out in Proposition 3.12. Propo-sition 3.10 then proves the theorem for such ϕ (1) and ϕ (2) . The second step is to show that the resultholds for more general functions ϕ (1) and ϕ (2) ; this step is carried out in Proposition 3.14.For technical reasons, and to prove more general results, we consider a more general operator. Definition.
Suppose that ϕ (1) and ϕ (2) are as described above, that χ (1) and χ (2) are the L -normalised characteristic functions of the unit balls in H ν and R , and that β, γ ∈ [0 , f ∈ L ( H ν ),we define the generalised area function S F, area , ϕ ,β,γ ( f ) S F, area , ϕ ,β,γ ( f )( g ) := (cid:18)Z Z R + × R + (cid:12)(cid:12) f ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s (cid:12)(cid:12) ∗ (1) χ (1) βr ∗ (2) χ (2) γs ( g ) drr dss (cid:19) / (3.12)for all g ∈ H ν , where we interpret χ (1)0 and χ (2)0 as Dirac delta distributions on H ν and R . Proposition 3.12.
Suppose that
M > D/ and N ≥ ; that ˜ ϕ (1) ∈ C M +2 ( H ν ) is radial and supportedin B (1) ( o, and ˜ ϕ (2) ∈ C N +2 ( R ) is even and supported in B (2) (0 , ; and that β, γ ∈ [0 , . Then Z ( S ∗ ) c S F, area , ϕ ,β,γ ( a R )( g ) dg ≤ C ( M, N, ν, ϕ ) ρ / , M − D/ ( R, S ) | R | / k a R k L ( H ν ) (3.13) for all particles a R such that supp( b R ) ⊆ R ∗ and all S ∈ R such that R ⊆ S with ρ / , M − D/ ( R, S ) defined as in (3.9) . Here, C ( M, N, ν, ϕ ) . M,N,ν (cid:13)(cid:13)(cid:13) ˜ ϕ (1) (cid:13)(cid:13)(cid:13) C M +2 ( H ν ) (cid:13)(cid:13)(cid:13) ˜ ϕ (2) (cid:13)(cid:13)(cid:13) C M +2 ( R ) . Proof.
We begin with a summary of the argument: given a particle a R such that a R = ∆ M (1) ∆ N (2) b R ,where supp( b R ) ⊆ R ∗ , we first translate and dilate a R so that R ∗ ⊆ T ( o, , h ), where h ≥
1, and S ∗ = T ( o, r ∗ , h ∗ ); next we simplify what has to be proved by splitting the sublinear operator S F, area , ϕ ,β,γ intofive pieces, which we treat separately; for several of these, we decompose ( S ∗ ) c into annular regionsand bound the L norm of the part of S F, area , ϕ ,β,γ ( a R ) under consideration in each of the regionsusing the Cauchy–Schwarz inequality. We now provide the details. In this proof, we write ϕ r,s for ϕ (1) r ∗ (2) ϕ (2) s and χ r,s for χ (1) r ∗ (2) χ (2) s . 38he estimate (3.13) is invariant under translations and normalised dilations, so there is no loss ofgenerality in supposing that R ⊆ R ∗ = T ( o, , h ) , where h ≥
1; then | R | h | R ∗ | h h . Moreover, wemay suppose that S ∗ = T ( o, r ∗ , h ∗ ) , where r ∗ ≥ h ∗ ≥ h . We observe that S F, area , ϕ ,β,γ ( a R ) h X j =1 S jF, area , ϕ ,β,γ ( a R ) , where S jF, area , ϕ ,β,γ ( a R )( g ) = (cid:18)Z Z T j (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs ( g ) drr dss (cid:19) / , and the regions T j , which partition R + × R + into five, are defined by T = { ( r, s ) : 0 < r ≤ , < s ≤ h } , T = { ( r, s ) : 0 < r ≤ , h < s < ∞} T = { ( r, s ) : 1 < r ≤ h / , r < s ≤ h } , T = { ( r, s ) : h / < r < ∞ , < s ≤ r } T = { ( r, s ) : 1 < r < ∞ , max { r , h } < s < ∞} . The key to our estimation of these terms is the observation thatsupp (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs ⊆ T ( o, , h ) · T ( o, r, s ) · T ( o, βr, γs ) ⊆ T ( o, r + 1 , s + h ) . (3.14)First, (3.14) implies that the support of S F, area , ϕ ( a R ) is so small that it does not meet ( S ∗ ) c andso no estimation is needed; nevertheless, we note that (cid:13)(cid:13) S F, area , ϕ ,β,γ ( a R ) (cid:13)(cid:13) L ( H ν ) . | R | / k a R k L ( H ν ) by the Cauchy–Schwarz inequality and the L -boundedness of the Lusin–Littlewood–Paley area oper-ator.Second, to treat S F, area , ϕ ,β,γ ( a R ), that is, to estimate Z ( S ∗ ) c (cid:18)Z Z [ h, ∞ ) × (0 , (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs ( g ) drr dss (cid:19) / dg, we define V k = T ( o, , × k h ) when k ∈ N ; then | V k | h k h , and supp (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs ⊆ V k when s ≤ k h , from (3.14). Moreover, a R ∗ (1) ϕ r,s = ∆ M (1) ∆ N (2) b R ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s = ∆ M (1) b R ∗ (1) ϕ (1) r ∗ (2) ∆ N (2) ( ϕ (2) s )= s − N ∆ M (1) b R ∗ (1) ϕ (1) r ∗ (2) (∆ N (2) ϕ (2) ) s . From these considerations, Z H ν \ V (cid:18)Z Z [ h, ∞ ) × (0 , (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs ( g ′ ) drr dss (cid:19) / dg ′ = X k ∈ Z + Z V k \ V k − (cid:18)Z Z [ h, ∞ ) × (0 , (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs ( g ′ ) drr dss (cid:19) / dg ′ = X k ∈ Z + Z V k \ V k − (cid:18)Z Z [2 k − h, ∞ ) × (0 , (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs ( g ′ ) drr dss (cid:19) / dg ′ ≤ X k ∈ Z + | V k \ V k − | / Z V k \ V k − Z Z [2 k − h, ∞ ) × (0 , (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs ( g ′ ) drr dss dg ′ ! / X k ∈ Z + | V k | / Z Z Z H ν × [2 k − h, ∞ ) × (0 , (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs ( g ′ ) drr dss dg ′ ! / = X k ∈ Z + | V k | / Z Z [2 k − h, ∞ ) × (0 , (cid:13)(cid:13) a R ∗ (1) ϕ r,s ( g ) (cid:13)(cid:13) L ( H ν ) drr dss ! / = X k ∈ Z + | V k | / Z Z [2 k − h, ∞ ) × (0 , s − N (cid:13)(cid:13)(cid:13) ∆ M (1) b R ∗ (1) ϕ (1) r ∗ (2) (∆ N (2) ϕ (2) ) s (cid:13)(cid:13)(cid:13) L ( H ν ) drr dss ! / ≤ X k ∈ Z + | V k | / (cid:13)(cid:13)(cid:13) ∆ N (2) ϕ (2) (cid:13)(cid:13)(cid:13) L ( R ) Z Z [2 k − h, ∞ ) × (0 , s − N (cid:13)(cid:13)(cid:13) ∆ M (1) b R ∗ (1) ϕ (1) r (cid:13)(cid:13)(cid:13) L ( H ν ) drr dss ! / = X k ∈ Z + | V k | / (cid:13)(cid:13)(cid:13) ∆ N (2) ϕ (2) (cid:13)(cid:13)(cid:13) L ( R ) Z [2 k − h, ∞ ) s − N dss ! / Z (0 , (cid:13)(cid:13)(cid:13) ∆ M (1) b R ∗ (1) ϕ (1) r (cid:13)(cid:13)(cid:13) L ( H ν ) drr ! / . ϕ ,N X k ∈ Z + | V k | / Nk h N (cid:13)(cid:13)(cid:13) ∆ M (1) b R (cid:13)(cid:13)(cid:13) L ( H ν ) . N X k ∈ Z + k/ h / kN k a R k L ( H ν ) h | R | / k a R k L ( H ν ) ;we need N ≥ S ∗ ∩ supp S F, area , ϕ ,β,γ ( a R ) ⊆ V k when k ≥ k ∗ := ⌈ log ((( r ∗ ) + h ∗ − / h ) ⌉ , it follows that Z ( S ∗ ) c (cid:18)Z Z [ h, ∞ ) × (0 , (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs ( g ′ ) drr dss (cid:19) / dg ′ ≤ ∞ X k = k ∗ − Z V k \ V k − (cid:18)Z Z [ h, ∞ ) × (0 , (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs ( g ′ ) drr dss (cid:19) / dg ′ . ϕ ,N ∞ X k = k ∗ − k/ h / kN k a R k L ( H ν ) . N (cid:18) h ( r ∗ ) + h ∗ − (cid:19) N − / | R | / k a R k L ( H ν ) h N (cid:18) height( R )height( S ) (cid:19) N − / | R | / k a R k L ( H ν ) . Third, we deal with S F, area , ϕ ,β,γ ( a R ), that is, with (cid:18)Z h / Z h (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs ( g ) dss drr (cid:19) / . We define U j to be the set { ( z, t ) ∈ H ν : | z | ≤ × j , | t | ≤ h } when j ∈ N ; then | U j | . νj h for all j . By (3.14), supp (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs ⊆ U j if 1 ≤ r < j and 0 < s ≤ h . Further, a R ∗ (1) ϕ r,s = ∆ M (1) ∆ N (2) b R ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s = ∆ N (2) b R ∗ (1) ∆ M (1) ( ϕ (1) r ) ∗ (2) ϕ (2) s = r − M ∆ N (2) b R ∗ (1) (∆ M (1) ϕ (1) ) r ∗ (2) ϕ (2) s . Z H ν \ U (cid:18)Z h / Z h (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs ( g ′ ) dss drr (cid:19) / dg ′ = ⌊ log ( h ) ⌋ +1 X j =1 Z U j \ U j − (cid:18)Z h / j − Z h (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs ( g ′ ) dss drr (cid:19) / dg ′ ≤ ⌊ log ( h ) ⌋ +1 X j =1 | U j \ U j − | / Z U j \ U j − Z h / j − Z h (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs ( g ′ ) dss drr dg ′ ! / ≤ X j ∈ Z + | U j | / (cid:18)Z H ν Z ∞ j − Z h (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs ( g ′ ) dss drr dg ′ (cid:19) / = X j ∈ Z + | U j | / (cid:18)Z ∞ j − Z h (cid:13)(cid:13)(cid:13) ∆ N (2) b R ∗ (1) (∆ M (1) ϕ (1) ) r ∗ (2) ϕ (2) s (cid:13)(cid:13)(cid:13) L ( H ν ) dss drr M +1 (cid:19) / . ϕ X j ∈ Z + | U j | / (cid:18)Z ∞ j − (cid:13)(cid:13)(cid:13) ∆ N (2) b R ∗ (1) (∆ M (1) ϕ (1) ) r (cid:13)(cid:13)(cid:13) L ( H ν ) drr M +1 (cid:19) / ≤ X j ∈ Z + | U j | / (cid:18)Z ∞ j − (cid:13)(cid:13)(cid:13) ∆ N (2) b R (cid:13)(cid:13)(cid:13) L ( H ν ) (cid:13)(cid:13)(cid:13) (∆ M (1) ϕ (1) ) r (cid:13)(cid:13)(cid:13) L ( H ν ) drr M +1 (cid:19) / . M X j ∈ Z + | U j | / (cid:13)(cid:13)(cid:13) ∆ N (2) b R (cid:13)(cid:13)(cid:13) L ( H ν ) (cid:13)(cid:13)(cid:13) ∆ M (1) ϕ (1) (cid:13)(cid:13)(cid:13) L ( H ν ) − jM . ϕ k a R k L ( H ν ) X j ∈ Z + jn | R | / − jM ≃ M | R | / k a R k L ( H ν ) ;we need M > ν/ S ∗ ∩ supp S F, area , ϕ ,β,γ ( a R ) ⊆ U j when j ≥ j ∗ := ⌊ log ( r ∗ / ⌋ , it follows that Z ( S ∗ ) c (cid:18)Z h / Z h (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs ( g ′ ) dss drr (cid:19) / dg ′ ≤ ∞ X j = j ∗ | U j | / (cid:18)Z H ν Z ∞ j − Z h (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs ( g ′ ) dss drr dg ′ (cid:19) / ≃ M, ϕ (cid:18) width( R )width( S ) (cid:19) M − ν | R | / k a R k L ( H ν ) . Fourth, to treat S F, area , ϕ ,β,γ ( a R ), that is, to estimate Z ( S ∗ ) c (cid:18)Z ∞ Z r (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ r,s ( g ) dss drr (cid:19) / dg, we define U j = T ( o, j +1 + 1 , j +1 + h ) when j ∈ N ; then | U j | . Dj + 2 νj h .By (3.14), if r < j and s ≤ r , then supp (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs ⊆ U j . Further, by hypothesis, ϕ (2) = ∆ (2) ˜ ϕ (2) , whence a R ∗ (1) ϕ r,s = ∆ M (1) ∆ N (2) b R ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s = ∆ M (1) ∆ N (2) b R ∗ (1) ϕ (1) r ∗ (2) (∆ (2) ˜ ϕ (2) ) s = s ∆ M (1) ∆ N (2) b R ∗ (1) ϕ (1) r ∗ (2) ∆ (2) ( ˜ ϕ (2) s )= s ∆ N (2) b R ∗ (1) ∆ M (1) ∆ (2) ( ϕ (1) r ) ∗ (2) ˜ ϕ (2) s = r − M − s ∆ N (2) b R ∗ (1) (∆ M (1) ∆ (2) ϕ (1) ) r ∗ (2) ˜ ϕ (2) s
41o that (cid:13)(cid:13) a R ∗ (1) ϕ r,s (cid:13)(cid:13) L ( H ν ) is bounded by r − M − s (cid:13)(cid:13)(cid:13) ∆ N (2) b R (cid:13)(cid:13)(cid:13) L ( H ν ) (cid:13)(cid:13)(cid:13) ∆ M (1) ϕ (1) (cid:13)(cid:13)(cid:13) L ( H ν ) (cid:13)(cid:13)(cid:13) ˜ ϕ (2) (cid:13)(cid:13)(cid:13) L ( R ) . ϕ ,M r − M − s k a R k L ( H ν ) . Matters are now arranged such that Z H ν \ U (cid:18)Z ∞ Z r (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs ( g ′ ) dss drr (cid:19) / dg ′ = X j ∈ Z + Z U j \ U j − (cid:18)Z ∞ j − Z r (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs ( g ′ ) dss drr (cid:19) / dg ′ ≤ X j ∈ Z + | U j \ U j − | / Z U j \ U j − Z ∞ j − Z r (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs ( g ′ ) dss drr dg ′ ! / ≤ X j ∈ Z + | U j | / Z H ν Z ∞ j − Z r (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs ( g ′ ) dss drr dg ′ ! / = X j ∈ Z + | U j | / Z ∞ j − Z r (cid:13)(cid:13) a R ∗ (1) ϕ r,s (cid:13)(cid:13) L ( H ν ) dss drr ! / . ϕ X j ∈ Z + | U j | / Z ∞ j − Z r r − − M s k a R k L ( H ν ) dss drr ! / ≃ M k a R k L ( H ν ) X j ∈ Z + (2 Dj + 2 νj h ) / − Mj h | R | / k a R k L ( H ν ) ;we need M > D/ S ∗ ∩ supp S F, area , ϕ ,β,γ ( a R ) ⊆ U j when j ≥ j ∗ , where j ∗ is the smallest integer j such that r ∗ ≤ j +1 +1 and ( r ∗ ) + h ∗ ≤ (2 j +1 +1) +2 j +1 + h ,it follows that Z ( S ∗ ) c (cid:18)Z ∞ Z r (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs ( g ′ ) dss drr (cid:19) / dg ′ . (cid:18) width( R )width( S ) (cid:19) M − D/ | R | / k a R k L ( H ν ) . Fifth, to estimate the term S F, area , ϕ ,β,γ ( a R ), that is, (cid:18)Z ∞ Z ∞ max { r ,h } (cid:12)(cid:12) a R ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s (cid:12)(cid:12) ∗ (1) χ (1) βr ∗ (2) χ (2) γs ( g ) dss drr (cid:19) / , we use a simple pointwise estimate that follows from (3.14), the normalisation of χ βr,γs , the com-mutativity of convolution on radial functions and distributions, and the Cauchy–Schwarz inequality,namely, (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs ≤ (cid:13)(cid:13)(cid:13)(cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs (cid:13)(cid:13)(cid:13) L ∞ ( H ν ) T ( o, r +1 , s + h ) ≤ (cid:13)(cid:13)(cid:13)(cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) (cid:13)(cid:13)(cid:13) L ∞ ( H ν ) T ( o, r +1 , s + h ) = (cid:13)(cid:13)(cid:13) b R ∗ (1) ∆ M (1) ∆ N (2) ϕ r,s (cid:13)(cid:13)(cid:13) L ∞ ( H ν ) T ( o, r +1 , s + h ) ≤ k b R k L ( H ν ) (cid:13)(cid:13)(cid:13) ∆ M (1) ∆ N (2) ϕ r,s (cid:13)(cid:13)(cid:13) L ( H ν ) T ( o, r +1 , s + h ) . (3.15)Further, we see that∆ M (1) ∆ N (2) ϕ r,s ( z, t ) = ∆ M (1) ( ϕ (1) r ) ∗ (1) ∆ N (2) ( ϕ (2) s )( z, t ) = r − M s − N (∆ M (1) ϕ (1) ) r ∗ (1) (∆ N (2) ϕ (2) ) s ( z, t )42 r − M s − N Z R (∆ M (1) ϕ (1) ) r ( z, t ′ )(∆ N (2) ϕ (2) ) s ( t − t ′ ) dt ′ ;the integrand vanishes off an interval of length O ( r ) because ( ϕ (1) ) r is supported in B (1) ( o, r ), so (cid:12)(cid:12)(cid:12) ∆ M (1) ∆ N (2) ϕ r,s ( z, t ) (cid:12)(cid:12)(cid:12) . r − M s − N (cid:13)(cid:13)(cid:13) (∆ M (1) ϕ (1) ) r (cid:13)(cid:13)(cid:13) L ∞ ( H ν ) (cid:13)(cid:13)(cid:13) (∆ N (2) ϕ (2) ) s (cid:13)(cid:13)(cid:13) L ∞ ( R ) = (cid:13)(cid:13)(cid:13) ∆ M (1) ϕ (1) (cid:13)(cid:13)(cid:13) L ∞ ( H ν ) (cid:13)(cid:13)(cid:13) ∆ N (2) ϕ (2) (cid:13)(cid:13)(cid:13) L ∞ ( R ) r − M − D s − N − ;since moreover ∆ M (1) ∆ N (2) ϕ r,s is supported in T ( o, r, s ), we deduce that (cid:13)(cid:13)(cid:13) ∆ M (1) ∆ N (2) ϕ r,s (cid:13)(cid:13)(cid:13) L ( H ν ) . ϕ ,M,N r − M − D s − N − | T ( o, r, s ) | / . r − M − ν s − N − / . (3.16)Moreover, from Corollary 3.2, k b R k L ( H ν ) . h N k a R k L ( H ν ) . Combining this with the estimates (3.15)and (3.16), we deduce that (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs . r − M − ν s − N − h N k a R k L ( H ν ) T ( o, r +1 , s + h ) . (3.17)When r ≥ s ≥ max { r , h } , and ( z, t ) ∈ T ( o, r + 1 , s + h ), | z | ≤ r + 1 ≤ r and | t | ≤ s + h + (2 r + 1) ≤ s. (3.18)It follows that (cid:18)Z ∞ Z ∞ max { r ,h } (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs ( z, t ) dss drr (cid:19) / ≤ Z ∞ Z ∞ max { r ,h } (cid:13)(cid:13)(cid:13)(cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ βr,γs (cid:13)(cid:13)(cid:13) L ∞ ( H ν ) T ( o, r +1 , s + h ) ( z, t ) dss drr ! / . ϕ,M,N h N k a R k L ( H ν ) Z ∞ max { , | z | / } Z ∞ max { r ,h, | t | / } r M +2 ν s N +1 dss drr ! / = h N k a R k L ( H ν ) N + 1 Z ∞ max { , | z | / } { r , h, | t | / } N +1 r M +2 ν drr ! / . M,N h N k a R k L ( H ν ) { , | z | / } M + ν max { , | z | / , h, | t | / } N +1 / . M,N k a R k L ( H ν ) h N max { , | z | / } M + ν max { , | z | / , h, | t | / } N +1 / for all ( z, t ) ∈ H ν . It follows immediately that (cid:13)(cid:13)(cid:13) S F, area , ϕ ,β,γ ( a R ) (cid:13)(cid:13)(cid:13) L ( H ν ) . | R | / k a R k L ( H ν ) providedthat M > ν/ N ≥
1. Further, Z ( S ∗ ) c S F, area , ϕ ,β,γ ( a R )( g ) dg . M,N,ϕ k a R k L ( H ν ) Z C ν { , | z | / } M + ν dz Z | t |≥ h ∗ h N ( | t | / N +1 / dt + k a R k L ( H ν ) Z | z |≥ r ∗ { , | z | / } M + ν dz Z R h N max { h, | t | / } N +1 / dt . M,N (cid:18) height( R )height( S ) (cid:19) N − / | R | / k a R k L ( H ν ) + (cid:18) width( R )width( S ) (cid:19) M − ν | R | / k a R k L ( H ν ) . | R | / in our estimates are dominated by ρ N − / , M − D/ ( R, S ), asdefined in (3.9). Further, examination of the argument shows that the implicit constants depend on ϕ through products such as (cid:13)(cid:13) ϕ (1) (cid:13)(cid:13) L ( H ν ) (cid:13)(cid:13)(cid:13) ∆ N (2) ϕ (2) (cid:13)(cid:13)(cid:13) L ∞ ( R ) , all of which may be controlled by the product (cid:13)(cid:13) ˜ ϕ (1) (cid:13)(cid:13) C M +2 ( H ν ) (cid:13)(cid:13) ˜ ϕ (2) (cid:13)(cid:13) C N +2 ( R ) because of the support restriction. Corollary 3.13.
Let S F, area , ϕ ,β,γ be as in Proposition 3.12. Then S F, area , ϕ ,β,γ ( f ) ∈ L ( H ν ) , and kS F, area , ϕ , , ( f ) k L ( H ν ) . k f k H F, atom ( H ν ) for all f ∈ H F, atom ( H ν ) ∩ L ( H ν ) . Hence S F, area , ϕ ,β,γ extends uniquely by density and continuity to amapping from H F, atom ( H ν ) to H F, area ( H ν ) .Proof. This follows from Proposition 3.10.We now consider what happens when the functions ϕ (1) and ϕ (2) in the definition of the Lusin–Littlewood–Paley area function do not have compact support. As in the statement of Theorem 3.11,we consider a Poisson-bounded pair of functions ˜ ϕ (1) on H ν and ˜ ϕ (2) on R , that is, functions such that (cid:12)(cid:12)(cid:12) D ˜ ϕ (1 ,r ) ( g ) (cid:12)(cid:12)(cid:12) . ( k g k + 1) − D − − m ∀ g ∈ H ν for all differential operators D that are products of m vector fields, each chosen from X , . . . , X ν , and (cid:12)(cid:12)(cid:12) T n ˜ ϕ (2 ,s ) ( t ) (cid:12)(cid:12)(cid:12) . ( | t | + 1) − − n ∀ t ∈ R . We define ϕ (1) = ∆ (1) ˜ ϕ (1) and ϕ (2) = ∆ (2) ˜ ϕ (2) . As before, we write ϕ r,s = ϕ (1) r ∗ (1) ϕ (2) s , where ϕ (1) r and ϕ (2) s are L -normalised dilates, and define χ r,s similarly. Proposition 3.14.
Suppose that the functions ϕ (1) and ϕ (2) are as defined above and S F, area , ϕ ,β,γ ( f ) is defined by (3.12) . Then S F, area , ϕ , , ( f ) ∈ L ( H ν ) , and kS F, area , ϕ , , ( f ) k L ( H ν ) . k f k H F, atom ( H ν ) ∀ f ∈ H F, atom ( H ν ) ∩ L ( H ν ) . Proof.
It is easy to construct smooth functions η ( k ) on R , where k ∈ N , such that P k ∈ N η ( k ) = 1, η (0) ( t ) = 0 unless | t | ≤ η (1) ( t ) = 0 unless 1 / ≤ | t | ≤
2, and η ( k ) = η (1) ( · / k − ) when k ≥
2. Takean even function ϕ (2) on R . Then we may write ϕ (2) = X k ∈ N ϕ (2) η ( k ) = X k ∈ N ( ϕ (2 ,k ) ) k , where ϕ (2 ,k ) is the normalised dilate ( ϕ (2) η ( k ) ) − k , which is supported in B (2) (0 , X k ∈ N (cid:13)(cid:13)(cid:13) ϕ (2 ,k ) (cid:13)(cid:13)(cid:13) C N +2 ( R ) = (cid:13)(cid:13)(cid:13) ( ϕ (2) η (0) ) (cid:13)(cid:13)(cid:13) C N +2 ( R ) + X k ∈ Z + Nk (cid:13)(cid:13)(cid:13) ( ϕ (2) η ( k ) ) (cid:13)(cid:13)(cid:13) C N +2 ( R ) < ∞ , since we may write each derivative T m ( ϕ (2) η ( k ) ) as a finite weighted sum of terms T n ϕ (2) T m − n η ( k ) ,and (cid:12)(cid:12)(cid:12) T n ϕ (2) ( t ) T m − n η ( k ) ( t ) (cid:12)(cid:12)(cid:12) . (1 + | t | ) − − n − ( m − n ) k supp( η ( k ) ) ( t ) . − k − mk for all t ∈ R . Similarly, given a Poisson-bounded function ϕ (1) , we may write ϕ (1) = P j ∈ N ( ϕ (1 ,j ) ) j , where ϕ (1 ,j ) is supported in B (1) ( o,
1) and P j ∈ N (cid:13)(cid:13) ϕ (1 ,j ) (cid:13)(cid:13) C M +2 ( H ν ) < ∞ . To do this, one uses cut-offfunctions that depend on a smooth norm on H ν , such as the Kor´anyi norm, and homogeneous dilations.44uppose that β, γ ≥
1. By a change of variables and the previous proposition, Z H ν (cid:18)Z Z R + × R + (cid:12)(cid:12) f ∗ (1) ϕ βr,γs (cid:12)(cid:12) ∗ (1) χ r,s ( g ) drr dss (cid:19) / dg = Z H ν (cid:18)Z Z R + × R + (cid:12)(cid:12) f ∗ (1) ϕ r,s (cid:12)(cid:12) ∗ (1) χ β − r,γ − s ( g ) drr dss (cid:19) / dg . M,N (cid:13)(cid:13)(cid:13) ϕ (1) (cid:13)(cid:13)(cid:13) C M +2 ( H ν ) (cid:13)(cid:13)(cid:13) ϕ (2) (cid:13)(cid:13)(cid:13) C N +2 ( R ) k f k H F, atom ( H ν ) . (3.19)We conclude by taking functions ϕ (1) and ϕ (2) as in the statement of the proposition, decomposingthem as sums P j ∈ N ( ϕ (1 ,j ) ) j and P k ∈ N ( ϕ (2 ,k ) ) k , and applying the estimate (3.19): kS F, area , ϕ , , ( f ) k L ( H ν ) = Z H ν Z Z R + × R + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X j,k ∈ N f ∗ (1) ( ϕ (1 ,j ) ) j r ∗ (2) ( ϕ (2 ,k ) ) k s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∗ (1) χ r,s ( g ) drr dss / dg ≤ X j,k ∈ N Z H ν (cid:18)Z Z R + × R + (cid:12)(cid:12)(cid:12) f ∗ (1) ( ϕ (1 ,j ) ) j r ∗ (2) ( ϕ (2 ,k ) ) k s (cid:12)(cid:12)(cid:12) ∗ (1) χ r,s ( g ) drr dss (cid:19) / dg . M,N X j,k ∈ N (cid:13)(cid:13)(cid:13) ϕ (1 ,j ) (cid:13)(cid:13)(cid:13) C M +2 ( H ν ) (cid:13)(cid:13)(cid:13) ϕ (2 ,k ) (cid:13)(cid:13)(cid:13) C N +2 ( R ) k f k H F, atom ( H ν ) . k f k H F, atom ( H ν ) , as required.For the next corollary, recall that the letters h and p are used for the heat and Poisson kernels. Corollary 3.15.
Suppose that ϕ (1) and ϕ (2) are either ∆ (1) h (1)1 and ∆ (2) h (2)1 , or ∆ (1) p (1)1 and ∆ (2) p (2)1 on H ν and on R , and that S F, area , ϕ ( f ) is defined by (4.2) . Then S F, area , ϕ ( f ) ∈ L ( H ν ) , and kS F, area , ϕ ( f ) k L ( H ν ) ≤ k f k H F, atom ( H ν ) ∀ f ∈ H F, atom ( H ν ) ∩ L ( H ν ) . Hence S F, area , ϕ extends continuously to a unique mapping from H F, atom ( H ν ) to H F, area ( H ν ) .Proof. This result follows from the proposition above and our estimates for the heat and Poissonkernels in Section 2.5. H F, atom ( H ν ) ⊆ H F, cts ( H ν ) In this section, we consider the continuous square function Hardy space. We begin by recalling thedefinition.
Definition.
Suppose that ϕ (1) and ϕ (2) are as at the beginning of Section 3.4. For f ∈ L ( H ν ), wedefine the continuous Littlewood–Paley square function S F, cts , ϕ ( f ) associated to ϕ (1) and ϕ (2) by S F, cts , ϕ ( f )( g ) := (cid:18)Z Z R + × R + (cid:12)(cid:12) f ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s ( g ) (cid:12)(cid:12) drr dss (cid:19) / for all g ∈ H ν . We define the continuous square function Hardy space H F, cts , ϕ ( H ν ), often abbreviatedto H F, cts ( H ν ), to be the set of all f ∈ L ( H ν ) for which kS F, cts , ϕ ( f ) k L ( H ν ) < ∞ , with norms k f k H F, cts , ϕ ( H ν ) := kS F, cts , ϕ ( f ) k L ( H ν ) . heorem 3.16. Suppose that the pair ϕ of radial functions ϕ (1) and ϕ (2) on H ν and R is Poisson-bounded, as in Definition 2.20. Then the operator S F, cts , ϕ is bounded from H F, atom ( H ν ) to L ( H ν ) .Proof. This result follows from Proposition 3.12, with the parameters β and γ there taken to be 0,combined with an almost identical version of the proof of Proposition 3.14. H F, atom ( H ν ) ⊆ H F, dis ( H ν ) In this section, we consider the discrete square function Hardy space, whose definition we now recall.
Definition.
Suppose that ϕ (1) and ϕ (2) are as at the beginning of Section 3.4. For f ∈ L ( H ν ), wedefine the discrete Littlewood–Paley square function S F, dis , ϕ ( f ) associated to ϕ (1) and ϕ (2) by S F, dis , ϕ ( f )( g ) := (cid:18) X ( m,n ) ∈ Z × Z (cid:12)(cid:12) f ∗ (1) ϕ (1)2 m ∗ (2) ϕ (2)2 n ( g ) (cid:12)(cid:12) (cid:19) / for all g ∈ H ν . We define the discrete square function Hardy space H F, dis , ϕ ( H ν ), often abbreviated to H F, dis ( H ν ), to be the set of all f ∈ L ( H ν ) for which kS F, dis , ϕ ( f ) k L ( H ν ) < ∞ , with norm k f k H F, dis , ϕ ( H ν ) := kS F, dis , ϕ ( f ) k L ( H ν ) . By replacing integrals by sums in the arguments of the previous section, we may show that thediscrete operator S F, dis , ϕ is bounded from H F, atom ( H ν ) to L ( H ν ). H F, atom ( H ν ) ⊆ H F, gmax ( H ν ) We recall the key points of Definition 2.20. A family F ( H ν ) of pairs of functions ( ϕ (1) , ϕ (2) ), where ϕ (1) : H ν → C and ϕ (2) : R → C , is said to be Poisson-bounded if there are constants C m and C n forall m ∈ N and all n ∈ N such that (cid:12)(cid:12)(cid:12) D ϕ (1) ( g ) (cid:12)(cid:12)(cid:12) ≤ C m (1 + k g k ) m + D/ / ∀ g ∈ H ν for all differential operators D that are products of m vector fields, each chosen from X , . . . , X ν , and (cid:12)(cid:12)(cid:12) T e ϕ (2) ( t ) (cid:12)(cid:12)(cid:12) ≤ C n (1 + k t k ) n +1 ∀ t ∈ R for all pairs ( ϕ (1) , ϕ (2) ) ∈ F ( H ν ). As usual, we write ϕ r,s for the convolution ϕ (1) r ∗ (1) ϕ (2) s , where ϕ (1) r and ϕ (2) s denote the normalised dilates of ϕ (1) and ϕ (2) .Given such a Poisson-bounded family, the grand maximal function M F, gmax ( f ) of f ∈ L ( H ν ) isdefined by M F, gmax ( f )( g ) = sup ϕ ∈ F ( H ν ) sup r,s ∈ R + (cid:12)(cid:12) f ∗ (1) ϕ r,s ( g ) (cid:12)(cid:12) ∀ g ∈ H ν . Definition 3.17.
The space H F, gmax ( H ν ) is defined to be the set of all f ∈ L ( H ν ) such that M F, gmax ( f ) ∈ L ( H ν ), with norm k f k H F, gmax ( H ν ) := kM F, gmax ( f ) k L ( H ν ) . (3.20)46e do not show the dependence of this space on the choice of family F ( H ν ), though evidently thisis important. Theorem 3.18.
Suppose that
M > D/ , N ≥ and κ > . Then H F, atom ( H ν ) ⊆ H F, gmax ( H ν ) , and k f k H F, gmax ( H ν ) . M,N k f k H F, atom ( H ν ) ∀ f ∈ H F, atom ( H ν ) . Proof.
The proof involves two steps: first we suppose that ϕ (1) and ϕ (2) have compact support andshow that there exists δ ∈ R + such that, for all particles a R for which b R is supported in R ∗ , Z ( S ∗ ) c |M F, gmax ( a R )( g ) | dg . ρ δ ,δ ( R, S ) | R | / k a R k L ( H ν ) , where S is any adapted rectangle that contains R ; this step is carried out in Proposition 3.12. Propo-sition 3.10 then proves the theorem for such ϕ (1) and ϕ (2) . The second step is to show that the resultholds for more general functions ϕ (1) and ϕ (2) ; this step is carried out in Proposition 3.14. Proposition 3.19.
Suppose that
M > D/ and N ≥ , that ϕ ∈ F ( H ν ) and that ϕ (1) and ϕ (2) aresupported in the balls B (1) ( o, and B (2) (0 , . Then Z ( S ∗ ) c M F, gmax ( a R )( g ) dg . M,N ρ / , M − D/ ( R, S ) | R | / k a R k L ( H ν ) (3.21) for all particles a R such that supp( b R ) ⊆ R ∗ and all S ∈ R such that R ⊆ S . Here, ρ / , M − D/ ( R, S ) ,as defined in (3.9) .Proof. We begin with a summary of the argument: given a particle a R such that a R = ∆ M (1) ∆ N (2) b R ,where supp( b R ) ⊆ R ∗ , we first translate and dilate a R so that R ∗ ⊆ T ( o, , h ), where h ≥
1, and S ∗ = T ( o, r ∗ , h ∗ ); next we simplify what has to be proved by splitting the sublinear operator S F, area , ϕ into five pieces, which we treat separately; for several of these, we decompose ( S ∗ ) c into annular regionsand bound the L norm of the part of M F, gmax ( a R ) under consideration in each of the regions usingthe Cauchy–Schwarz inequality. We now provide the details. Once again, we write ϕ r,s for ϕ (1) r ∗ (2) ϕ (2) s and χ r,s for χ (1) r ∗ (2) χ (2) s .The estimate (3.13) is invariant under translations and normalised dilations, so there is no loss ofgenerality in supposing that R ⊆ R ∗ = T ( o, , h ); then | R | h | R ∗ | h h . Moreover, we may supposethat S ∗ = T ( o, r ∗ , h ∗ ) , where r ∗ ≥ h ∗ ≥ h . We observe that M F, gmax ( a R ) h max j =1 ,..., M jF, gmax ( a R ) , where M jF, gmax ( a R )( g ) = sup ϕ ∈ F ( H ν ) sup ( r,s ) ∈ T j (cid:12)(cid:12) a R ∗ (1) ϕ r,s ( g ) (cid:12)(cid:12) , and the regions T j , which partition R + × R + into five, are defined by T = { ( r, s ) : 0 < r ≤ , < s ≤ h } , T = { ( r, s ) : 0 < r ≤ , h < s < ∞} T = { ( r, s ) : 1 < r ≤ h / , r < s ≤ h } , T = { ( r, s ) : h / < r < ∞ , < s ≤ r } T = { ( r, s ) : 1 < r < ∞ , max { r , h } < s < ∞} . The key to our estimation of these terms is the observation thatsupp (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ⊆ T ( o, , h ) · T ( o, r, s ) = T ( o, r + 1 , s + h ) . (3.22)47irst, the support of M F, gmax ( a R ) is so small that it does not meet ( S ∗ ) c and so no estimation isneeded; nevertheless, we note that (cid:13)(cid:13)(cid:13) M F, gmax ( a R ) (cid:13)(cid:13)(cid:13) L ( H ν ) . | R | / k a R k L ( H ν ) by the Cauchy–Schwarzinequality and the L -boundedness of M F, gmax , which follows from its control by the flag maximalfunction M F (as in Definition 2.7).Second, to treat M F, gmax ( a R ), that is, to estimate Z ( S ∗ ) c sup ϕ ∈ F ( H ν ) sup r ∈ (0 , sup s ∈ [ h, ∞ ) (cid:12)(cid:12) a R ∗ (1) ϕ r,s ( g ) (cid:12)(cid:12) dg, (3.23)we define V k = T ( o, , × k h ) when k ∈ N ; then | V k | h k h , and supp (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ⊆ V k when s ≤ k h , from (3.22). Moreover, a R ∗ (1) ϕ r,s = ∆ M (1) ∆ N (2) b R ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s = ∆ M (1) b R ∗ (1) ϕ (1) r ∗ (2) ∆ N (2) ( ϕ (2) s )= s − N ∆ M (1) b R ∗ (1) ϕ (1) r ∗ (2) (∆ N (2) ϕ (2) ) s . From these considerations, Z H ν \ V sup ϕ ∈ F ( H ν ) sup r ∈ (0 , sup s ∈ [ h, ∞ ) (cid:12)(cid:12) a R ∗ (1) ϕ r,s ( g ) (cid:12)(cid:12) dg = X k ∈ Z + Z V k \ V k − sup ϕ ∈ F ( H ν ) sup r ∈ (0 , sup s ∈ [2 k − h, ∞ ) (cid:12)(cid:12) a R ∗ (1) ϕ r,s ( g ) (cid:12)(cid:12) dg ≤ X k ∈ Z + | V k \ V k − | / Z V k \ V k − sup ϕ ∈ F ( H ν ) sup r ∈ (0 , sup s ∈ [2 k − h, ∞ ) (cid:12)(cid:12) a R ∗ (1) ϕ r,s ( g ) (cid:12)(cid:12) dg ′ ! / ≤ X k ∈ Z + | V k | / (2 k − h ) − N Z H ν sup ϕ ∈ F ( H ν ) sup r ∈ (0 , sup s ∈ [2 k − h, ∞ ) (cid:12)(cid:12)(cid:12) ∆ M (1) b R ∗ (1) ϕ (1) r ∗ (2) (∆ N (2) ϕ (2) ) s ( g ) (cid:12)(cid:12)(cid:12) dg ! / ≤ X k ∈ Z + | V k | / (2 k − h ) − N sup ϕ ∈ F ( H ν ) (cid:18)(cid:13)(cid:13)(cid:13) ϕ (1) (cid:13)(cid:13)(cid:13) L ∞ ( H ν ) (cid:13)(cid:13)(cid:13) ∆ N (2) ϕ (2) (cid:13)(cid:13)(cid:13) L ∞ ( R ) (cid:19)(cid:18)Z H ν sup r ∈ R + sup s ∈ R + (cid:12)(cid:12)(cid:12) ∆ M (1) b R ∗ (1) χ (1) r ∗ (2) χ (2) s ( g ) (cid:12)(cid:12)(cid:12) dg (cid:19) / . F X k ∈ Z + | V k | / (2 k − h ) − N (cid:13)(cid:13)(cid:13) M s (∆ M (1) b R f ) (cid:13)(cid:13)(cid:13) L ( H ν ) . N h / h − N (cid:13)(cid:13)(cid:13) ∆ M (1) b R f (cid:13)(cid:13)(cid:13) L ( H ν ) . | R | / k a R k L ( H ν ) ;we need N ≥ S ∗ ∩ supp M F, gmax ( a R ) ⊆ V k when k ≥ k ∗ := ⌈ log ((( r ∗ ) + h ∗ − / h ) ⌉ , it follows that the left-hand side of (3.23) is bounded by ∞ X k = k ∗ − Z V k \ V k − sup ϕ ∈ F ( H ν ) sup r ∈ (0 , sup s ∈ [2 k − h, ∞ ) (cid:12)(cid:12) a R ∗ (1) ϕ r,s ( g ) (cid:12)(cid:12) dg . F ∞ X k = k ∗ − | V k | / (2 k − h ) − N (cid:13)(cid:13)(cid:13) M s (∆ M (1) b R f ) (cid:13)(cid:13)(cid:13) L ( H ν ) . N (cid:18) h ( r ∗ ) + h ∗ − (cid:19) N − / | R | / k a R k L ( H ν ) N (cid:18) height( R )height( S ) (cid:19) N − / | R | / k a R k L ( H ν ) . Third, we deal with M F, gmax ( a R ), that is, with Z ( S ∗ ) c sup ϕ ∈ F ( H ν ) sup r ∈ (1 ,h / ] sup s ∈ (0 ,h ] (cid:12)(cid:12) a R ∗ (1) ϕ r,s ( g ) (cid:12)(cid:12) dg. We define U j as U j := { ( z, t ) ∈ H ν : | z | ≤ j +1 , | t | ≤ h } when j ∈ N ; then | U j | . νj h for all j . By(3.22), supp( a R ∗ (1) ϕ r,s ) ⊆ U j if 1 ≤ r < j and 0 < s ≤ h . Further, a R ∗ (1) ϕ r,s = ∆ M (1) ∆ N (2) b R ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s = ∆ N (2) b R ∗ (1) ∆ M (1) ( ϕ (1) r ) ∗ (2) ϕ (2) s = r − M ∆ N (2) b R ∗ (1) (∆ M (1) ϕ (1) ) r ∗ (2) ϕ (2) s . Matters are now arranged such that Z H ν \ U sup ϕ ∈ F ( H ν ) sup r ∈ (1 ,h / ] sup s ∈ (0 ,h ] (cid:12)(cid:12) a R ∗ (1) ϕ r,s ( g ) (cid:12)(cid:12) dg = ⌊ log ( h ) ⌋ +1 X j =1 Z U j \ U j − sup ϕ ∈ F ( H ν ) sup r ∈ (2 j − ,h / ] sup s ∈ (0 ,h ] (cid:12)(cid:12) a R ∗ (1) ϕ r,s ( g ) (cid:12)(cid:12) dg ≤ ⌊ log ( h ) ⌋ +1 X j =1 | U j \ U j − | / Z U j \ U j − sup ϕ ∈ F ( H ν ) sup r ∈ (2 j − ,h / ] sup s ∈ (0 ,h ] (cid:12)(cid:12) a R ∗ (1) ϕ r,s ( g ) (cid:12)(cid:12) dg ! / ≤ ⌊ log ( h ) ⌋ +1 X j =1 | U j | / − M ( j − Z U j \ U j − sup ϕ ∈ F ( H ν ) sup r ∈ (2 j − ,h / ] sup s ∈ (0 ,h ] (cid:12)(cid:12)(cid:12) ∆ N (2) b R ∗ (1) (∆ M (1) ϕ (1) ) r ∗ (2) ϕ (2) s ( g ) (cid:12)(cid:12)(cid:12) dg ! / ≤ ⌊ log ( h ) ⌋ +1 X j =1 | U j | / − M ( j − sup ϕ ∈ F ( H ν ) (cid:18)(cid:13)(cid:13)(cid:13) ∆ M (1) ϕ (1) (cid:13)(cid:13)(cid:13) L ∞ ( H ν ) (cid:13)(cid:13)(cid:13) ϕ (2) (cid:13)(cid:13)(cid:13) L ∞ ( R ) (cid:19)(cid:18)Z H ν sup r ∈ R + sup s ∈ R + (cid:12)(cid:12)(cid:12) ∆ N (2) b R ∗ (1) χ (1) r ∗ (2) χ (2) s ( g ) (cid:12)(cid:12)(cid:12) dg (cid:19) / . F ⌊ log ( h ) ⌋ +1 X j =1 | U j | / − M ( j − (cid:13)(cid:13)(cid:13) M s (∆ N (2) b R ) (cid:13)(cid:13)(cid:13) L ( H ν ) . M | h | / k a R k L ( H ν ) ≃ | R | / k a R k L ( H ν ) ;we need M > ν/ S ∗ ∩ supp M F, gmax ( a R ) ⊆ U j when j ≥ j ∗ := ⌊ log ( r ∗ / ⌋ , it follows that Z ( S ∗ ) c sup ϕ ∈ F ( H ν ) sup r ∈ (1 ,h / ] sup s ∈ (0 ,h ] (cid:12)(cid:12) a R ∗ (1) ϕ r,s ( g ) (cid:12)(cid:12) dg . F ∞ X j = j ∗ | U j | / − M ( j − (cid:13)(cid:13)(cid:13) M s (∆ N (2) b R ) (cid:13)(cid:13)(cid:13) L ( H ν ) ≃ M (cid:18) width( R )width( S ) (cid:19) M − ν | R | / k a R k L ( H ν ) . Fourth, to treat M F, gmax ( a R ), that is, to estimate Z ( S ∗ ) c sup ϕ ∈ F ( H ν ) sup r ∈ (1 , ∞ ] sup s ∈ (0 ,r ] (cid:12)(cid:12) a R ∗ (1) ϕ r,s ( g ) (cid:12)(cid:12) dg,
49e define U j = T ( o, j + 1 , j + h ) when j ∈ N ; then | U j | . Dj + 2 νj h .By (3.22), if r < j and s ≤ r , then supp( a R ∗ (1) ϕ r,s ) ⊆ U j . Further, as before, a R ∗ (1) ϕ r,s = ∆ M (1) ∆ N (2) b R ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s = r − M ∆ N (2) b R ∗ (1) (∆ M (1) ϕ (1) ) r ∗ (2) ϕ (2) s . Matters are now arranged such that Z H ν \ U sup ϕ ∈ F ( H ν ) sup r ∈ (1 , ∞ ] sup s ∈ (0 ,r ] (cid:12)(cid:12) a R ∗ (1) ϕ r,s ( g ) (cid:12)(cid:12) dg = X j ∈ Z + Z U j \ U j − sup ϕ ∈ F ( H ν ) sup r ∈ (2 j − , ∞ ] sup s ∈ (0 ,r ] (cid:12)(cid:12) a R ∗ (1) ϕ r,s ( g ) (cid:12)(cid:12) dg ≤ X j ∈ Z + | U j \ U j − | / Z U j \ U j − sup ϕ ∈ F ( H ν ) sup r ∈ (2 j − , ∞ ] sup s ∈ (0 ,r ] (cid:12)(cid:12) a R ∗ (1) ϕ r,s ( g ) (cid:12)(cid:12) dg ! / ≤ X j ∈ Z + | U j | / − M ( j − Z U j \ U j − sup ϕ ∈ F ( H ν ) sup r ∈ (2 j − , ∞ ] sup s ∈ (0 ,r ] (cid:12)(cid:12)(cid:12) ∆ N (2) b R ∗ (1) (∆ M (1) ϕ (1) ) r ∗ (2) ϕ (2) s ( g ) (cid:12)(cid:12)(cid:12) dg ! / ≤ X j ∈ Z + | U j | / − M ( j − sup ϕ ∈ F ( H ν ) (cid:18)(cid:13)(cid:13)(cid:13) ∆ M (1) ϕ (1) (cid:13)(cid:13)(cid:13) L ∞ ) H ν ) (cid:13)(cid:13)(cid:13) ϕ (2) (cid:13)(cid:13)(cid:13) L ∞ ( R ) (cid:19)(cid:18)Z H ν sup r ∈ R + sup s ∈ R + (cid:12)(cid:12)(cid:12) ∆ N (2) b R ∗ (1) χ (1) r ∗ (2) χ (2) s ( g ) (cid:12)(cid:12)(cid:12) dg (cid:19) / . F ∞ X j =1 | U j | / − M ( j − (cid:13)(cid:13)(cid:13) M s (∆ N (2) b R ) (cid:13)(cid:13)(cid:13) L ( H ν ) . M | h | / k a R k L ( H ν ) ≃ | R | / k a R k L ( H ν ) ;we need M > D/ S ∗ ∩ supp M F, gmax ( a R ) ⊆ U j when j ≥ j ∗ ,where j ∗ is the smallest integer j such that r ∗ ≤ j +1 + 1 and ( r ∗ ) + h ∗ ≤ (2 j +1 + 1) + 2 j +1 + h , itfollows that Z ( S ∗ ) c sup ϕ ∈ F ( H ν ) sup r ∈ (1 , ∞ ] sup s ∈ (0 ,r ] (cid:12)(cid:12) a R ∗ (1) ϕ r,s ( g ) (cid:12)(cid:12) dg . (cid:18) width( R )width( S ) (cid:19) M − D/ | R | / k a R k L ( H ν ) . Fifth, to estimate the term M F, gmax ( a R ), that is,sup ϕ ∈ F ( H ν ) sup r ∈ (1 , ∞ ] sup s ∈ (max { r ,h } , ∞ ] (cid:12)(cid:12) a R ∗ (1) ϕ r,s ( g ) (cid:12)(cid:12) , we use the pointwise estimate (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) ≤ (cid:13)(cid:13) a R ∗ (1) ϕ r,s (cid:13)(cid:13) L ∞ ( H ν ) T ( o,r +1 ,s + h ) = (cid:13)(cid:13)(cid:13) b R ∗ (1) ∆ M (1) ∆ N (2) ϕ r,s (cid:13)(cid:13)(cid:13) L ∞ ( H ν ) T ( o,r +1 ,s + h ) ≤ k b R k L ( H ν ) (cid:13)(cid:13)(cid:13) ∆ M (1) ∆ N (2) ϕ r,s (cid:13)(cid:13)(cid:13) L ( H ν ) T ( o,r +1 ,s + h ) . (3.24)Arguing very much as in the proof of (3.17), we see that (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) . C ( M, N, ϕ ) r − M − ν s − N − / h N k a R k L ( H ν ) T ( o,r +1 ,s + h ) C ( M, N, ϕ ) = (cid:13)(cid:13)(cid:13) ∆ M (1) ϕ (1) (cid:13)(cid:13)(cid:13) L ∞ ( H ν ) (cid:13)(cid:13)(cid:13) ∆ N (2) ϕ (2) (cid:13)(cid:13)(cid:13) L ∞ ( R ) . Further, we see that∆ M (1) ∆ N (2) ϕ r,s ( z, t ) = ∆ M (1) ( ϕ (1) r ) ∗ (1) ∆ N (2) ( ϕ (2) s )( z, t ) = r − M s − N (∆ M (1) ϕ (1) ) r ∗ (1) (∆ N (2) ϕ (2) ) s ( z, t )= r − M s − N Z R (∆ M (1) ϕ (1) ) r ( z, t ′ )(∆ N (2) ϕ (2) ) s ( t − t ′ ) dt ′ ;the integrand vanishes off an interval of length O ( r ) because ( ϕ (1) ) r is supported in B (1) ( o, r ), so (cid:12)(cid:12)(cid:12) ∆ M (1) ∆ N (2) ϕ r,s ( z, t ) (cid:12)(cid:12)(cid:12) . r − M s − N (cid:13)(cid:13)(cid:13) (∆ M (1) ϕ (1) ) r (cid:13)(cid:13)(cid:13) L ∞ ( H ν ) (cid:13)(cid:13)(cid:13) (∆ N (2) ϕ (2) ) s (cid:13)(cid:13)(cid:13) L ∞ ( R ) . ϕ,M,N r − M − D s − N − ;since moreover ∆ M (1) ∆ N (2) ϕ r,s is supported in T ( o, r, s ), we deduce that (cid:13)(cid:13)(cid:13) ∆ M (1) ∆ N (2) ϕ r,s (cid:13)(cid:13)(cid:13) L ( H ν ) . ϕ,M,N r − M − D s − N − | T ( o, r, s ) | / . r − M − ν s − N − / . (3.25)Moreover, from Corollary 3.2, k b R k L ( H ν ) . h N k a R k L ( H ν ) . Combining this with the estimates in(3.24) and (3.25), we deduce that (cid:12)(cid:12) a R ∗ (1) ϕ r,s (cid:12)(cid:12) . r − M − ν s − N − / h N k a R k L ( H ν ) T ( o,r +1 ,s + h ) . (3.26)When r ≥ s ≥ max { r , h } , and ( z, t ) ∈ T ( o, r + 1 , s + h ), | z | ≤ r + 1 ≤ r and | t | ≤ s + h + ( r + 1) ≤ s. (3.27)Consequently,sup ϕ ∈ F ( H ν ) sup r ∈ (1 , ∞ ] sup s ∈ (max { r ,h } , ∞ ] (cid:12)(cid:12) a R ∗ (1) ϕ r,s ( z, t ) (cid:12)(cid:12) . F sup r ∈ (1 , ∞ ] sup s ∈ (max { r ,h } , ∞ ] r − M − ν s − N − / h N k a R k L ( H ν ) T ( o,r +1 ,s + h ) ( z, t ) . M,N { , | z | / } M + ν h N max { h, | t | /g } M + ν k a R k L ( H ν ) It follows immediately that (cid:13)(cid:13)(cid:13) M F, gmax ( a R ) (cid:13)(cid:13)(cid:13) L ( H ν ) . | R | / k a R k L ( H ν ) as long as M > ν/ N ≥ Z ( S ∗ ) c M F, gmax ( a R )( g ) dg . M,N,ϕ k a R k L ( H ν ) Z C ν { , | z | / } M + ν dz Z | t |≥ h ∗ h N ( | t | / N +1 / dt + k a R k L ( H ν ) Z | z |≥ r ∗ { , | z | / } M + ν dz Z R h N max { h, | t | / } N +1 / dt . M,N (cid:18) height( R )height( S ) (cid:19) N − / | R | / k a R k L ( H ν ) + (cid:18) width( R )width( S ) (cid:19) M − ν | R | / k a R k L ( H ν ) . The quotients that precede | R | / in our estimates are dominated by ρ N − / , M − D/ ( R, S ), asdefined in (3.9).Further, examination of the argument shows that ϕ comes into the constants in the form ofproducts such as (cid:13)(cid:13) ϕ (1) (cid:13)(cid:13) L ( H ν ) (cid:13)(cid:13)(cid:13) ∆ N (2) ϕ (2) (cid:13)(cid:13)(cid:13) L ∞ ( R ) , which are all dominated by (cid:13)(cid:13) ϕ (1) (cid:13)(cid:13) C M ( H ν ) · (cid:13)(cid:13) ϕ (2) (cid:13)(cid:13) C N ( R ) because of the support restriction. 51e now consider what happens when the functions ϕ (1) and ϕ (2) in the definition of the grandmaximal function are not supported in the unit balls of H ν and R . Proposition 3.20.
Suppose that the functions ϕ (1) and ϕ (2) are Poisson-bounded, as in Definition2.20, and that M F, gmax ( f ) is defined by (3.17) . Then M F, gmax ( f ) ∈ L ( H ν ) , and kM F, gmax ( f ) k L ( H ν ) . k f k H F, atom ( H ν ) ∀ f ∈ H F, atom ( H ν ) ∩ L ( H ν ) . Proof.
The proof of this proposition is essentially the same as the proof of Proposition 3.14.We recall that p (1) r and p (2) s denote the Poisson kernels on H ν and on R (here r, s ∈ R + ), and, for f ∈ L ( H ν ), the Poisson integral u of f is given by u ( g, r, s ) := f ∗ (1) p (1) r ∗ (2) p (2) s ( g ) ∀ g ∈ H ν . The nontangential maximal function of f ∈ L ( H ν ) is defined by u ∗ ( g ) = sup ( g ′ ,r,s ) ∈ Γ β ( g ) | u ( g ′ , r, s ) | , and the radial maximal function of f ∈ L ( H ν ) is defined by u + ( g ) = sup r,s ∈ R + | u ( g ′ , r, s ) | . Corollary 3.21.
Suppose that f ∈ H F, atom ( H ν ) . The the radial maximal function u + and the non-tangential maximal function U ∗ are in L ( H ν ) , and (cid:13)(cid:13) u + (cid:13)(cid:13) L ( H ν ) ≤ k u ∗ k L ( H ν ) . k f k H F, atom ( H ν ) ∀ f ∈ H F, atom ( H ν ) . Proof.
It is obvious that the grand maximal function dominates the radial and nontangential maximalfunctions.
In this section, we consider two types of singular integral operator acting on the atomic Hardy space.We denote by H the Hilbert transform in the central variable, that is, H f ( z, t ′ ) = Z R f ( z, t ′ − t ) dtt ∀ ( z, t ′ ) ∈ H ν . We shall be interested mainly in the case where f ∈ L ( H ν ), in which case one may interpret H as aconvolution operator acting on the functions f ( z, · ) for almost all z ∈ C ν . The second type of operatorwas discussed in Section 2.6; we write K for the operator of right convolution with a smooth singularintegral kernel that is homogeneous of degree − D . Theorem 3.22.
Suppose that H and K are as discussed above. Then KH and K are bounded operatorsfrom H F, atom ( H ν ) to L ( H ν ) . roof. We are going to apply Proposition 3.10, and need to estimate Z ( S ∗ ) c |K a R ( g ) | dg, where a R is a particle associated to an adapted rectangle R and S is an adapted rectangle that contains R . We also need to estimate a similar integral with KH a R in place of K a R . We begin by consideringthe action of H , and reducing the examination of the second integral to the examination of the first.Suppose that ˜ a ∈ L ( R ) and ˜ a = ∆ (2) ˜ b , where ˜ b ∈ L ( R ) and is supported in [ − h, h ]. Using astandard partition of unity argument, we may write H ˜ a as a sum P n ∈ Z a ∗ (2) k n , where k n is an oddkernel, which vanishes outside [ − n +1 h, − n − h ] ∪ [2 n − h, n +1 h ] and k n ( · ) = k (2 n · ) for all n . Thekernel P n = −∞ k n is a truncated version of the Hilbert kernel, so the associated convolution operatoris bounded on L ( R ); the bound is independent of h . Hence for ˜ a and ˜ b as above, we may view H ˜ a as the sum of one function ˜ a ∗ (2) ( P n = −∞ k n ), which has controlled L ( R )-norm and is supported in[ − h, h ], and functions ˜ a ∗ (2) k n , where n >
0, which are supported in [ − (2 n +1 + 1) h, (2 n +1 + 1) h ].Since ˜ a ∗ (2) k n = ˜ b ∗ (2) ∆ (2) k n = ∆ (2) (˜ b ∗ (2) k n ) , and (cid:13)(cid:13) ∆ (2) k n (cid:13)(cid:13) L ( H ν ) . − n h − , it follows that (cid:12)(cid:12) supp(˜ a ∗ (1) k n ) (cid:12)(cid:12) / (cid:13)(cid:13) ˜ a ∗ (1) k n (cid:13)(cid:13) L ( H ν ) . n h / (cid:13)(cid:13) ∆ (2) k n (cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ b (cid:13)(cid:13)(cid:13) L ( H ν ) . − n h / k ˜ a k L ( H ν ) (3.28)by Lemma 2.23. In short, ˜ a ∗ (1) k n is a multiple of a particle with support in [ − n +2 h, n +2 h ].Suppose now that a R is a normalised particle supported in an enlargement R ∗ of an adaptedrectangle of base q and height h . We may write H a R = a R ∗ (2) X n = −∞ k n ! + X n ∈ Z + a R ∗ (2) k n =: a R, + X n ∈ Z + a R,n , which is a weighted sum of normalised particles supported in tubes R ∗ · [ − n +1 h, n +1 h ]. The left-hand summand a R, has similar L ( H ν )-norm and support size to the particle a R , and poses noproblems. The other terms a R,n are particles with smaller L ( H ν )-norms and larger support sizes,and some thought is required to deal with these. To do this, we write R n for the smallest adaptedrectangle that contains R · [ − n +1 κ − h, n +1 κ − h ]; it has the same base as R , but a larger height, and R ∗ · [ − n +1 h, n +1 h ] ⊆ R ∗ n . Then the summands a R,n may be associated with the adapted rectangles R n , and from (3.28), | R n | / k a R,n k L ( H ν ) . − n | R | / k a R k L ( H ν ) . Suppose now that we can show that there exists δ , δ ∈ R + such that Z ( S ∗ ) c |K a R ( g ) | dg . M,N,δ ρ δ ,δ ( R, S ) | R | / k a R k L ( H ν ) (3.29)for all particles a R associated to R ∈ R and all S ∈ R with ρ δ ,δ ( R, S ), as defined in (3.9). It willthen follow that K is bounded from H F, atom ( H ν ) to L ( H ν ) by Proposition 3.10. Further, from above,53 a R = P n ∈ N a R,n , whence Z ( S ∗ ) c |KH a R ( g ) | dg = Z ( S ∗ ) c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K X n ∈ N a R,n ( g ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dg ≤ X n ∈ N Z ( S ∗ ) c |K a R,n ( g ) | dg ≤ X n ∈ N ρ δ ,δ ( R n , S ) | R n | / k a R,n k L ( H ν ) . X n ∈ N ρρ δ ,δ ( R n , S )2 − n | R | / k a R k L ( H ν ) . ρ δ ,δ ( R, S ) | R | / k a R k L ( H ν ) . It remains to prove (3.29). To do this, we use a partition of unity to decompose the convolu-tion kernel k of the operator K as a sum P j ∈ N k j , where supp( k ) ⊆ B (1) ( o,
1) and supp( k j ) ⊆ B (1) ( o, j − ) \ B (1) ( o, j ) when j ≥
1; we may do this in such a way that convolution with k isbounded on L ( H ν ), and the L ( H ν )-norms of the derivatives of the kernels k j vanish rapidly as j increases. When we consider the convolutions a R ∗ (1) k j , we can pass a derivative of a R onto k j toobtain a convergent sum.We note that the argument of the proof above may easily be improved to show that K and KH send particles to H F, atom ( H ν )-functions. However, it appears to be nontrivial to then show that theseoperators also send atoms to H F, atom ( H ν )-functions. Nevertheless, this conclusion follows from ourlater characterisation of the Hardy space using Riesz transforms. In this section examine the Hardy space defined via area functions. We first examine the definition andproperties of the space, and then show that if f ∈ H F, area , ϕ ( H ν ), and ϕ (1) and ϕ (2) satisfy appropriatespectral conditions, then f ∈ H F, atom ( H ν ). We show that the space is independent of some of the paramenters used in its definition, and provethat the area operator is (when suitably normalised) an isometry on L ( H ν ).We begin by introducing three cones that we shall discuss. Definition 4.1.
Suppose that g ∈ H ν and β, γ ∈ R + . The cones Γ β,γ ( g ), Γ (1) β ( g ), and Γ (2) β ( g ) aredefined as follows: Γ β,γ ( g ) = { ( g ′ , r, s ) ∈ H ν × R + × R + : g ′ ∈ T ( g, βr, γs ) } , Γ (1) β ( g ) = { ( g ′ , r ) ∈ H ν × R + : g ′ ∈ g · B (1) ( o, βr ) } , Γ (2) γ ( g ′′ ) = { ( g ′′′ , s ) ∈ R × R + : g ′′′ ∈ g ′′ · B (2) (0 , γs ) } . The cone Γ β defined earlier (as in (1.1)) corresponds to the cone Γ β,β above.Next, we recall from Section 2.3 that S ∆ (1) ( H ν ) is the subspace of S ( H ν ) of convolution kernels ofoperators Φ (1) (∆ (1) ) where Φ (1) ∈ S ( R ), and S ∆ (2) ( R ) is the subspace of S ( R ) of convolution kernels54f operators Φ (2) (∆ (2) ) where Φ (2) ∈ S ( R ). Spectral theory provides an abundance of such functions˜ ϕ (1) and ˜ ϕ (2) , and in particular, we may also suppose that ˜ ϕ (1) and hence ϕ (1) , and ˜ ϕ (2) and hence ϕ (2) are compactly supported. Moreover, by multiplying ϕ (1) and ϕ (2) by appropriate scalars, we maysuppose that Z R + | Φ (1) ( ru ) | drr = Z R + | Φ (2) ( sv ) | dss = 1 ∀ u, v ∈ R + . (4.1)Finally, the normalised characteristic functions | B (1) ( o, r ) | − B (1) ( o,r ) and | B (2) (0 , s ) | − B (2) (0 ,s ) aredenoted by χ (1) r and χ (2) s . We indicate the pair of functions ϕ (1) and ϕ (2) by the symbol ϕ . Definition 4.2.
Suppose that ϕ (1) r , ϕ (2) s , χ (1) r and χ (2) s are as above, and that β, γ ∈ R + . For f ∈ L ( H ν ), we define the Lusin–Littlewood–Paley area function S F, area , ϕ ,β,γ ( f ) associated to ϕ (1) and ϕ (2) by S F, area , ϕ ,β,γ ( f )( g ) := (cid:18)Z Z R + × R + (cid:12)(cid:12) f ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s (cid:12)(cid:12) ∗ (1) χ (1) βr ∗ (2) χ (2) γs ( g ) drr dss (cid:19) / (4.2)for all g ∈ H ν , and we define the Hardy space H F, area , ϕ ,β,γ ( H ν ), usually abbreviated to H F, area ( H ν ),to be the set of all f ∈ L ( H ν ) for which kS F, area , ϕ ,β,γ ( f ) k L ( H ν ) < ∞ , with norm k f k H F, area , ϕ ,β,γ ( H ν ) := kS F, area , ϕ ,β,γ ( f ) k L ( H ν ) . (4.3)Important special cases of the Lusin–Littlewood–Paley area function include when ϕ (1) and ϕ (2) are the convolution kernels of the heat semigroup or the Poisson semigroup on H ν and R .There are several equivalent definitions of the area integral, as illustrated by the following lemma. Lemma 4.3.
The following integrals are equivalent: I β,γ := Z H ν (cid:18)Z Z R + × R + ( F ( · , r, s ) ∗ (1) χ (1) βr ∗ (2) χ (2) γs )( g ) drr dss (cid:19) / dg ;J β,γ := Z H ν Z Z Z Γ β,γ ( g ) F ( g ′ , r, s ) | T ( o, βr, γs ) | dg ′ drr dss ! / dg, for all functions F : H ν × R + × R + → [0 , + ∞ ) and all β, γ ∈ R + . The implicit constants are geometric.Proof. We claim that I β,γ h I β ′ ,γ ′ for all β , γ , β ′ and γ ′ , and then that I β,γ/ . J β,γ . I β,γ , whichimplies the stated result.First, Z Z R + × R + ( F ( · , r, s ) ∗ (1) χ (1) βr ∗ (2) χ (2) γs )( g ) drr dss = Z Z Z H ν × R + × R + F ( gg , r, s )( χ (1) βr ∗ (2) χ (2) γs )( g − ) dg drr dss = Cβ D γ Z Z Z Z H ν × R × R + × R + F ( gg g , r, s ) B (1) ( o,βr ) ( g ) B (2) (0 ,γs ) ( g ) dg dg drr D +1 dss , (4.4)where C depends on ν . The last expression may be rewritten as either Cβ D γ Z Z Γ (1) β ( o ) Z Z Γ (2) γ (0) F ( gg g , r, s ) dss dg ! drr D +1 dg or 55 β D γ Z Z Γ (2) γ (0) Z Z Γ (1) β ( o ) F ( gg g , r, s ) drr D +1 dg ! dss dg . Hence I β,γ is equivalent to either Cβ D γ Z H ν Z Z Γ (1) β ( o ) Z Z Γ (2) γ (0) F ( gg g , r, s ) dss dg ! drr D +1 dg ! / dg or Cβ D γ Z H ν Z Z Γ (2) γ (0) Z Z Γ (1) β ( o ) F ( gg g , r, s ) drr D +1 dg ! dss dg ! / dg. It may be argued, as in the classical case (see [57, pp. 125–126]) that changing the parameter β changes the first of these two expressions to an equivalent expression, and that changing the parameter γ changes the second to an equivalent expression (in this case, we need to write g as ( z, t ) and integratewith respect to first t and then z ). We conclude that changing β and γ changes I β,γ to an equivalentintegral.Finally, from Lemma 2.8, I β,γ/ . J β,γ . I β,γ , (4.5)and the lemma follows. Corollary 4.4.
The space H F, area , ϕ ,β,γ ( H ν ) coincides with the space of functions on H ν for which Z H ν (cid:18)Z Z Z Γ β ( g ) | T ( o, βr, γs ) | (cid:12)(cid:12) f ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s ( g ′ ) (cid:12)(cid:12) dg ′ drr dss (cid:19) / dg and the integral above is equivalent to the H F, area , ϕ ,β,γ ( H ν ) norm. Since the Hardy space H F, area , ϕ ,β,γ ( H ν ) does not depend on the parameters β and γ , henceforthwe take these to be 1 unless explicitly stated otherwise, and write H F, area , ϕ ( H ν ) or just H F, area ( H ν )for the space. Lemma 4.5.
There exists a positive geometric constant C P such that, if Ω is a subset of H ν of finitemeasure, and W = S g ∈ Ω Γ( g ) , then Ω ∗ (1) p r,s ( g ′ ) ≥ C P ∀ ( g ′ , r, s ) ∈ W. Proof.
By Lemma 2.21, for all β, γ ∈ R + , Ω ∗ (1) p r,s ( g ′ ) & Ω ∗ (1) χ αr,βs ( g ′ ) , ∀ ( g ′ , r, s ) ∈ H ν × R + × R + . The proof is complete.Finally, we show that the Lusin–Littlewood–Paley area operator is bounded on L ( H ν ). Proposition 4.6.
Suppose that ϕ (1) and ϕ (2) are as described before Definition 4.2. For all f ∈ L ( H ν ) , S F, area , ϕ ( f ) ∈ L ( H ν ) , and kS F, area , ϕ ( f ) k L ( H ν ) = k f k L ( H ν ) . Proof.
Since the functions χ (1) r and χ (2) s are normalised characteristic functions, for all f ∈ L ( H ν ), kS F, area , ϕ ( f ) k L ( H ν ) := Z Z Z H ν × R + × R + (cid:12)(cid:12)(cid:12) f ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s (cid:12)(cid:12)(cid:12) ∗ (1) χ (1) r ∗ (2) χ (2) s ( g ) drr dss dg = Z Z Z H ν × R + × R + (cid:12)(cid:12)(cid:12) f ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s ( g ) (cid:12)(cid:12)(cid:12) drr dss dg = k f k L ( H ν ) , as required. 56 .2 The inclusion H F, area ( H ν ) ⊆ H F, atom ( H ν ) Here we show that every f ∈ H F, area ( H ν ) has an atomic decomposition, and control the decompositionappropriately, provided ϕ (1) and ϕ (2) satisfy certain spectral conditions. To be more specific, we willshow the following result. Theorem 4.7.
Suppose that
M, N ≥ and that ψ (1) and ψ (2) satisfy . . . . There exists a constant C ,depending on M , N , ν , and ϕ , such that for all f ∈ H F, area , ψ ( H ν ) ∩ L ( H ν ) , there exist (1 , , M, N, atoms a j and numbers λ j , for all j ∈ N , such that f ∼ P λ j a j , where the sum converges in L ( H ν ) ,and k f k H F, atom ( H ν ) ≤ ∞ X j =0 | λ j | ≤ C k f k H F, area , ψ ( H ν ) . Hence, by continuity and density, H F, area , ψ ( H ν ) ⊆ H F, atom ( H ν ) .Proof. We denote by R the collection of all adapted rectangles, as in Section 2.2. We take theenlargement parameter κ (defined at the start of Section 3) to be 3, and then (2.13) implies that R · B (1) ( o, q ) · B (2) (0 , h ) ⊂ g · ¯ B (1) ( o, q/ · ¯ B (2) (0 , ( q + 4 h ) / · B (1) ( o, q ) · B (2) (0 , h )= g · B (1) ( o, q/ · B (2) (0 , ( q + 12 h ) / ⊆ B (1) ( o, κq/ · B (2) (0 , κ ( q + 4 h ) / R ∗ , when R ∈ R , width( R ) = q and height( R ) = h . By Lemma 2.9, for all open sets Ω and R ∈ R (Ω) R ∗ ⊆ (cid:26) g ∈ H ν : M F ( Ω )( g ) > ν κ D (5 ν + 2) (cid:27) . For each tile R in R , the tent T ( R ) over R is defined by T ( R ) := (cid:8) ( g, r, s ) ∈ H ν × R + × R + : g ∈ R, r ∈ ( q/ , q ] , s ∈ (0 , q ] (cid:9) , where q = width( R ), and for each adapted rectangle R in R that is not a tile, the tent T ( R ) over R is defined by T ( R ) := (cid:8) ( g, r, s ) ∈ H ν × R + × R + : g ∈ R, r ∈ ( q/ , q ] , s ∈ ( h/ , h ] (cid:9) , where q = width( R ) and h = height( R ). It is evident that H ν × R + × R + decomposes as a disjointunion: H ν × R + × R + = G ℓ ∈ Z R ∈ B ℓ T ( R );indeed, for ( g, r, s ) ∈ H ν × R + × R + , the coordinates r and s determine the dimensions and thecoordinate g determines the location of an adapted rectangle R such that g ∈ R .Take f ∈ H F, area ( H ν ) ∩ L ( H ν ). For each ℓ ∈ Z , we defineΩ ℓ := n g ∈ H ν : S F, area , ϕ ( f )( g ) > ℓ o , B ℓ := (cid:26) R ∈ R : | R ∗ ∩ Ω ℓ | > κ D | R ∗ | , | R ∗ ∩ Ω ℓ +1 | ≤ κ D | R ∗ | (cid:27) What conditions? Ω ℓ := (cid:26) g ∈ H ν : M F ( Ω ℓ )( g ) > ν κ D (5 ν + 2) (cid:27) . The second of these exotic-looking constants is chosen so that R ∗ ⊆ e Ω ℓ ; we leave it to the reader tocheck that the first ensures that, if R ∈ R and ( g, r, s ) ∈ T ( R ), then13 κ D | R ∗ | ≤ | T ( g, r, s ) | . (4.6)We take smooth even radial functions ˜ ψ (1) on H ν and ˜ ψ (2) on R such that supp ˜ ψ (1) ⊆ B (1) ( o, ψ (2) ⊆ B (2) (0 , ψ (1) = ∆ M (1) ˜ ψ (1) and ψ (2) = ∆ N (2) ˜ ψ (2) ; by rescaling ϕ (1) and ϕ (2) if necessary, and renormalising, we may and shall assume that Z R + f ∗ (1) ϕ (1) r ∗ (1) ψ (1) r drr = f and Z R + f ∗ (2) ϕ (2) s ∗ (2) ψ (2) s drr = f for all f ∈ L ( H ν ); here, as usual, ϕ (1) r denotes the normalised dilate of ϕ (1) , and so on. As before, weabbreviate ϕ (1) r ∗ (2) ϕ (2) s to ϕ r,s , and define ˜ ψ r,s and ψ r,s analogously. It follows from spectral theory,as in Corollary 2.16 and Remark 2.17, that for all f ∈ H F, area ( H ν ) ∩ L ( H ν ), f = Z Z R + × R + f ∗ (1) ϕ r,s ∗ (1) ψ r,s dss drr = Z H ν Z Z R + × R + f ∗ (1) ϕ r,s ( g ) g ψ r,s dss drr dg = X ℓ ∈ Z X R ∈B ℓ Z Z Z T ( R ) f ∗ (1) ϕ r,s ( g ) g ψ r,s drr dss dg = X ℓ ∈ Z λ ℓ X R ∈ B ℓ a ℓ,R , say, where a ℓ,R := 1 λ ℓ Z Z Z T ( R ) f ∗ (1) ϕ r,s ( g ) g ψ r,s dg drr dss and λ ℓ := (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) X R ∈ B ℓ Z Z R + × R + (cid:12)(cid:12) f ∗ (1) ϕ r,s ( · ) (cid:12)(cid:12) T ( R ) ( · ) drr dss (cid:19) / (cid:13)(cid:13)(cid:13)(cid:13) L ( H ν ) | e Ω ℓ | / . (If λ ℓ = 0, then all a ℓ,R are taken to be 0, as the integral involved in the definition of a ℓ,R vanishes.At the end of this proof we show that each λ ℓ is finite, and so these expressions make sense.)Fix ℓ ∈ Z ; we claim that P R ∈ B ℓ a ℓ,R is a geometric multiple of a flag atom. In light of Lemma 3.3,it suffices to prove that there exist functions b ℓ,R in Dom(∆ M (1) ∆ N (2) ) in L ( H ν ), for all R ∈ B ℓ , suchthat(A1) a ℓ,R = ∆ (1) M ∆ (2) N b R and supp b R ⊆ R ∗ ; and(A2) for all sign sequences σ : B ℓ → {± } , the sum P R ∈ B ℓ σ ℓ,R a ℓ,R converges in L ( H ν ), to a σ say,and k a σ k L ( H ν ) . | Ω | − / . Suppose that width( R ) = q and height( R ) = h . It is evident that for each ℓ ∈ Z and R ∈ B ℓ , a ℓ,R := ∆ M (1) ∆ N (2) b ℓ,R , where b ℓ,R := 1 λ ℓ Z Z Z T ( R ) f ∗ (1) ϕ r,s ( g ) g ˜ ψ r,s dg drr dss . (4.7)58y construction, when ( g, r, s ) ∈ T ( R ), supp g ˜ ψ r,s ⊆ g · T ( o, r, s ) ⊆ R · T ( o, q, h ) . Hencesupp b ℓ,R ⊆ R ∗ . (4.8)Next we take a sign sequence σ : B ℓ → {± } and estimate the L ( H ν )-norm of P R ∈ B ℓ σ ℓ,R a ℓ,R .For all smooth compactly supported h on H ν such that k h k L ( H ν ) ≤ (cid:12)(cid:12)(cid:12)Z H ν X R ∈ B ℓ σ ℓ,R a ℓ,R ( g ′ ) h ( g ′ ) dg ′ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ ℓ X R ∈ B ℓ σ ℓ,R Z Z Z T ( R ) Z H ν f ∗ (1) ϕ r,s ( g ) g ψ r,s ( g ′ ) h ( g ′ ) dg ′ dss drr dg (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ ℓ X R ∈ B ℓ σ ℓ,R Z Z Z T ( R ) h ∗ (1) ψ r,s ( g ) f ∗ (1) ϕ r,s ( g ) dss drr dg (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ ℓ X R ∈ B ℓ σ ℓ,R Z Z Z H ν × R + × R + h ∗ (1) ψ r,s ( g ) f ∗ (1) ϕ r,s ( g ) T ( R ) ( g, r, s ) dss drr dg (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ λ ℓ (cid:18) X R ∈ B ℓ Z Z Z H ν × R + × R + | h ∗ (1) ψ r,s ( g ) | T ( R ) ( g, r, s ) dss drr dg (cid:19) / (cid:18) X R ∈ B ℓ Z Z Z H ν × R + × R + (cid:12)(cid:12) f ∗ (1) ϕ r,s ( g ) (cid:12)(cid:12) T ( R ) ( g, r, s ) dss drr dg (cid:19) / ≤ λ ℓ (cid:18)Z Z Z H ν × R + × R + | h ∗ (1) ψ r,s ( g ) | dg (cid:19) / (cid:18) X R ∈ B ℓ Z Z Z H ν × R + × R + (cid:12)(cid:12) f ∗ (1) ϕ r,s ( g ) (cid:12)(cid:12) R ( g ) T d ( R ) ( m, n ) dg (cid:19) / . λ ℓ k h k L ( H ν ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) X R ∈ B ℓ Z Z R + × R + (cid:12)(cid:12) f ∗ (1) ϕ r,s ( · ) (cid:12)(cid:12) T ( R ) ( · , r, s ) dss drr (cid:19) / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( H ν ) ≤ | e Ω ℓ | − / , by the Cauchy–Schwarz inequality and the definition of λ ℓ , and so (cid:13)(cid:13)(cid:13)P R ∈ B ℓ σ ℓ,R a ℓ,R (cid:13)(cid:13)(cid:13) L ( H ν ) . | e Ω ℓ | − / .Hence a is a multiple of an atom, and the multiple depends only on M , N , ν , ϕ and ψ .Finally, we verify the convergence of the series P ℓ | λ ℓ | . To do this, we first fix ℓ ∈ Z , and observethat if R ∈ R and ( g ′ , r, s ) ∈ T ( R ), then, from (2.13), T ( g ′ , r, s ) ⊆ R ∗ ⊆ e Ω ℓ , whence (cid:12)(cid:12)(cid:12)e Ω ℓ ∩ T ( g ′ , r, s ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12) T ( g ′ , r, s ) (cid:12)(cid:12) = | T ( o, r, s ) | , while, by definition of B ℓ and (4.6), (cid:12)(cid:12) Ω ℓ +1 ∩ T ( g ′ , r, s ) (cid:12)(cid:12) ≤ | R ∗ ∩ Ω ℓ +1 | ≤ κ D | R ∗ | ≤ | T ( o, r, s ) | . It follows that (cid:12)(cid:12)(cid:12) ( e Ω ℓ \ Ω ℓ +1 ) ∩ T ( g ′ , r, s ) (cid:12)(cid:12)(cid:12) | T ( g ′ , r, s ) | ≥ . Now by the definitions of B ℓ , the Lusin–Littlewood–Paley area function, and Ω ℓ , X R ∈ B ℓ Z Z Z T ( R ) (cid:12)(cid:12) f ∗ (1) ϕ r,s ( g ′ ) (cid:12)(cid:12) dss drr dg ′ X R ∈ B ℓ Z Z Z T ( R ) (cid:12)(cid:12) f ∗ (1) ϕ r,s ( g ′ ) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ( e Ω ℓ \ Ω ℓ +1 ) ∩ T ( g ′ , r, s ) (cid:12)(cid:12)(cid:12) | T ( g ′ , r, s ) | drr dss dg ′ ≤ Z Z Z H ν × R + × R + (cid:12)(cid:12) f ∗ (1) ϕ r,s ( g ′ ) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ( e Ω ℓ \ Ω ℓ +1 ) ∩ T ( g ′ , r, s ) (cid:12)(cid:12)(cid:12) | T ( g ′ , r, s ) | drr dss dg ′ = 2 Z e Ω ℓ \ Ω ℓ +1 Z Z Z H ν × R + × R + (cid:12)(cid:12) f ∗ (1) ϕ r,s ( g ′ ) (cid:12)(cid:12) T ( o,r,s ) ( g ′ · g − ) | T ( o, r, s ) | drr dss dg ′ dg = 2 Z e Ω ℓ \ Ω ℓ +1 Z Z Z Γ( g ) (cid:12)(cid:12) f ∗ (1) ϕ r,s ( g ′ ) (cid:12)(cid:12) | T ( o, r, s ) | drr dss dg ′ dg = 2 Z e Ω ℓ \ Ω ℓ +1 |S F, area , ϕ ( f )( g ) | dg ≤ ℓ +3 | e Ω ℓ | . Hence X ℓ ∈ Z | λ ℓ | ≤ X ℓ ∈ Z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) X R ∈ B ℓ Z Z R + × R + (cid:12)(cid:12) f ∗ (1) ψ r,s ( · ) (cid:12)(cid:12) T ( R ) ( · , r, s ) drr dss (cid:19) / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( H ν ) (cid:12)(cid:12)(cid:12)e Ω ℓ (cid:12)(cid:12)(cid:12) / = X ℓ ∈ Z (cid:18) X R ∈ B ℓ Z Z Z T ( R ) (cid:12)(cid:12) f ∗ (1) ψ r,s ( g ′ ) (cid:12)(cid:12) dg ′ drr dss (cid:19) / | e Ω ℓ | / ≤ X ℓ ∈ Z ℓ +3 / | e Ω ℓ | . kS F, area , ψ ( f ) k L ( H ν ) = k f k H F, area , ψ ( H ν ) . This completes the proof of Theorem 4.7.We remark that the atomic decomposition P ℓ ∈ Z λ ℓ a ℓ in the proof above converges to f in the L ( H ν ) norm. Indeed, to show this, we only need to show that k P | ℓ | >L λ ℓ a ℓ k L ( H ν ) → L tendsto infinity. To see this, first note that (cid:12)(cid:12)(cid:12)(cid:10) X | ℓ | >L λ ℓ a ℓ , h (cid:11)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) X R ∈ B ℓ Z Z Z T ( R ) h ∗ (1) ψ r,s ( g ′ ) X | ℓ | >L λ ℓ a ℓ ∗ (1) ϕ r,s ( g ′ ) dg ′ drr dss (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z H ν (cid:18) X R ∈ B ℓ Z Z R + × R + (cid:12)(cid:12) h ∗ (1) ψ r,s ( g ′ ) (cid:12)(cid:12) T ( R ) ( g ′ , r, s ) drr dss (cid:19) / (cid:18) X R ∈ B ℓ Z Z R + × R + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | ℓ | >L λ ℓ a ℓ ∗ (1) ϕ r,s ( g ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ( R ) ( g ′ , r, s ) drr dss (cid:19) / dg ′ ≤ C k h k L ( H ν ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) X R ∈ B ℓ Z Z R + × R + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | ℓ | >L λ ℓ a ℓ ∗ (1) ϕ r,s ( g ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ( R ) ( g ′ , r, s ) drr dss (cid:19) / (cid:13)(cid:13)(cid:13)(cid:13) L ( H ν ) → L tends to ∞ , as the area integral operator is bounded on L ( H ν ). Since (cid:26) X | ℓ | >L λ ℓ a ℓ (cid:27) L ( H ν ) = sup h : k h k L H ν )=1 (cid:12)(cid:12)(cid:12)(cid:10) X | ℓ | >L λ ℓ a ℓ , h (cid:11)(cid:12)(cid:12)(cid:12) , P ℓ ∈ Z λ ℓ a ℓ converges to f in the L ( H ν ) norm. We could replace the integrals over r and s in the definition of the area function S F, area , ϕ (Definition4.2 by sums, taking r and s to be 2 j and 2 k , or more generally e αj and e βk .The proofs that S F, area , ϕ ( f ) ∈ L ( H ν ) if f ∈ H F, atom ( H ν ) in Section 3.4 and that f ∈ H F, atom ( H ν )if S F, area , ϕ ( f ) ∈ L ( H ν ) in Section 4.2 go through with minor modifications, and show that the atomicHardy space may also be characterised by a “discrete area function”. We spare the reader the details! We recall the definitions of the continuous and discrete Littlewood–Paley square functions. We supposethat the integers M and N satisfy M > D/ N ≥
1, and consider a pair ˜ ϕ of Poisson-boundedfunctions, as in Definition 2.20. We define ϕ (1) = ∆ (1) ˜ ϕ (1) and ϕ (2) = ∆ (2) ˜ ϕ (2) , and take ϕ (1) r to be r − D ϕ (1) ( δ /r · ) and ϕ (2) s to be s − ϕ (2) ( · /s ). Definition 5.1.
Suppose that ϕ (1) and ϕ (2) are as above. For f ∈ L ( H ν ), we define the continuousand discrete Littlewood–Paley square functions S F, cts , ϕ ( f ) and S F, dis , ϕ ( f ) associated to ϕ (1) and ϕ (2) by S F, cts , ϕ ( f )( g ) := (cid:18)Z Z R + × R + (cid:12)(cid:12) f ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s ( g ) (cid:12)(cid:12) drr dss (cid:19) / S F, dis , ϕ ( f )( g ) := (cid:18) X ( m,n ) ∈ Z × Z (cid:12)(cid:12) f ∗ (1) ϕ (1)2 m ∗ (2) ϕ (2)2 n ( g ) (cid:12)(cid:12) (cid:19) / for all g ∈ H ν . We define the square function Hardy spaces H F, cts , ϕ ( H ν ) and H F, dis , ϕ ( H ν ), often abbre-viated to H F, cts ( H ν ) and H F, dis ( H ν ), to be the set of all f ∈ L ( H ν ) for which kS F, cts , ϕ ( f ) k L ( H ν ) < ∞ or kS F, dis , ϕ ( f ) k L ( H ν ) < ∞ , with norms k f k H F, cts , ϕ ( H ν ) := kS F, cts , ϕ ( f ) k L ( H ν ) , k f k H F, dis , ϕ ( H ν ) := kS F, dis , ϕ ( f ) k L ( H ν ) . We showed in Section 3 that H F, atom ( H ν ) ⊆ H F, cts ( H ν ) and H F, atom ( H ν ) ⊆ H F, dis ( H ν ). Now weshow that H F, cts ( H ν ) ⊆ H F, area ( H ν ); coupled with the result of the last section that H F, area ( H ν ) ⊆ H F, atom ( H ν ), this completes the proof of the equivalence of these spaces. A key technical result on theway to showing that H F, cts ( H ν ) ⊆ H F, area ( H ν ) is that H F, cts , ϕ ( H ν ) = H F, cts , ψ ( H ν ) for suitable pairsof functions ϕ and ψ .Very similar arguments apply to the discrete square function space H F, dis ( H ν ), but we just sum-marise the changes that need to be made rather than go through all the details.Of course, for spaces such as H F, cts , ϕ ( H ν ) to be meaningful, we need some additional conditionson the pair of functions ϕ involved in the definition of the square function spaces. For instance if ϕ (1) = 0, then the spaces H F, cts ( H ν ) and H F, dis ( H ν ) are just L ( H ν ) with a trivial norm. H F, cts ( H ν ) ⊆ H F, area ( H ν ) A simple trick now enables us to pass from a square function to an area function.61 emma 5.2.
Suppose that ( ϕ (1) , ϕ (2) ) and ( ψ (1) , ψ (2) ) are as in Definition 5.1. Then kS F, area , ψ ( f ) k L ( H ν ) . kS F, cts , ϕ ( f ) k L ( H ν ) ∀ f ∈ L ( H ν ) . (5.1) Proof.
To begin with, we note that ( ϕ (1) , ϕ (2) ) and ( ψ (1) , ψ (2) ) given as in Definition 5.1 satisfy theproperties as in Definition def:ctssqfnphi. From the definition of S F, area , ψ ( f ), we have kS F, area , ψ ( f ) k L ( H ν ) = Z H ν (cid:18)Z Z R + × R + (cid:12)(cid:12) f ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s (cid:12)(cid:12) ∗ (1) χ (1) r ∗ (2) χ (2) s ( g ) drr dss (cid:19) / dg = Z H ν (cid:26) X j,k ∈ Z X R ∈ R ( N,j,k ) Z − j +1 − N − j − N Z − k +1 − N − k − N Z H ν χ (1) r ∗ (2) χ (2) s (˜ g − · g ) (cid:12)(cid:12) f ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s (˜ g ) (cid:12)(cid:12) d ˜ g drr dss χ R ( g ) (cid:27) / dg ≤ Z H ν (cid:26) X j,k ∈ Z X R ∈ R ( N,j,k ) Z − j +1 − N − j − N Z − k +1 − N − k − N Z H ν χ (1) r ∗ (2) χ (2) s (˜ g − · g ) sup g R ∈ CR (cid:12)(cid:12) f ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s ( g R ) (cid:12)(cid:12) d ˜ g drr dss χ R ( g ) (cid:27) / dg . Z H ν (cid:26) X j,k ∈ Z X R ∈ R ( N,j,k ) Z − j +1 − N − j − N Z − k +1 − N − k − N sup g R ∈ CR (cid:12)(cid:12) f ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s ( g R ) (cid:12)(cid:12) drr dss χ R ( g ) (cid:27) / dg, where the first inequality follows from the fact that for R ∈ R ( N, j, k ), g ∈ R , χ (1) r ∗ (2) χ (2) s (˜ g − · g ) = 0and for r ∈ (2 − j − N , − j +1 − N ), s ∈ (2 − k − N , − k +1 − N ), we see that there exists an absolute constant C such that ˜ g ∈ CR ; and second inequality follows from the fact that Z H ν χ (1) r ∗ (2) χ (2) s (˜ g − · g ) d ˜ g . . Then, by using the flag Plancherel–P´olya inequality, we deduce that kS F, area , ψ ( f ) k L ( H ν ) . Z H ν (cid:26) X j,k ∈ Z X R ∈ R ( N,j,k ) Z − j +1 − N − j − N Z − k +1 − N − k − N inf g R ∈ R (cid:12)(cid:12) f ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s ( g R ) (cid:12)(cid:12) drr dss χ R ( g ) (cid:27) / dg ≤ Z H ν (cid:18)Z Z R + × R + (cid:12)(cid:12) f ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s ( g ) (cid:12)(cid:12) drr dss (cid:19) / dg, which proves (5.1). H F, dis ( H ν ) ⊆ H F, atom ( H ν ) The arguments of the previous sections may be extendeded to show that the Hardy spaces H F, cts , ϕ ( H ν )and H F, cts , ψ ( H ν ) are equivalent, under appropriate conditions on the pairs of functions ϕ and ψ , whichare very similar to the conditions on the pairs of functions in Lemma 2.25.The argument of the previous section then implies that if f ∈ H F, dis ( H ν ), then f also belongs tothe “discrete area function” Hardy space, which we discussed (without defining precisely) in Section4.3, and explained why this space coincides with the atomic Hardy space.It follows that the discrete square function Hardy space coincides with the atomic Hardy space.62 Maximal function flag Hardy space
In this section, we characterise the flag Hardy space by maximal functions. We begin by recallingthree definitions.
Definition 6.1.
Fix β >
0. The nontangential maximal function of f ∈ L ( H ν ) is defined by u ∗ ( g ) = u ∗ ( f )( g ) = sup ( g ′ ,r,s ) ∈ Γ β ( g ) | u ( g ′ , r, s ) | . The space H F, nontan ( H ν ) is defined to be the linear space of all f ∈ L ( H ν ) such that k u ∗ k L ( H ν ) < ∞ ,with norm k f k H F, nontan ( H ν ) := k u ∗ ( f ) k L ( H ν ) . (6.1) Definition 6.2.
The radial maximal function of f ∈ L ( H ν ) is defined by u + ( g ) = u + ( f )( g ) = sup r,s ∈ R + | u ( g, r, s ) | . The space H F, radial ( H ν ) is defined to be the set of all f ∈ L ( H ν ) such that k u + k L ( H ν ) < ∞ , withnorm k f k H F, radial ( H ν ) := (cid:13)(cid:13) u + ( f ) (cid:13)(cid:13) L ( H ν ) . (6.2) Definition 6.3.
Let F ( H ν ) be the collection of all pairs ϕ of functions ϕ (1) on H ν and ϕ (2) on R suchthat (cid:12)(cid:12)(cid:12) ∆ M (1) ϕ (1) ( g ) (cid:12)(cid:12)(cid:12) ≤ k g k ) M +2 N + D/ / ∀ g ∈ H ν (cid:12)(cid:12)(cid:12) ∆ N (2) ϕ (2) ( t ) (cid:12)(cid:12)(cid:12) ≤ k t k ) M +1 ∀ t ∈ R . The grand maximal function M F, gmax ( f ) of f ∈ L ( H ν ) is defined by M F, gmax ( f )( g ) = sup ϕ ∈ F ( H ν ) sup r,s ∈ R + (cid:12)(cid:12) f ∗ (1) ϕ r,s ( g ) (cid:12)(cid:12) ∀ g ∈ H ν . The space H F, gmax ( H ν ) is defined to be the set of all f ∈ L ( H ν ) such that kM F, gmax ( f ) k L ( H ν ) < ∞ ,with norm k f k H F, gmax ( H ν ) := kM F, gmax ( f ) k L ( H ν ) . (6.3)In this definition, we write ϕ r,s for the convolution ϕ (1) r ∗ (1) ϕ (2) s , where ϕ (1) r and ϕ (2) s denote thenormalised dilates of ϕ (1) and ϕ (2) .In particular, the pairs of heat kernels ( h (1)1 , h (2)1 ) and pairs of Poisson kernels ( p (1)1 , p (2)1 ) and alltheir translates by elements of bounded subsets of H ν and of R belong to F ( H ν ), at least up tomultiples. It follows that the grand maximal function dominates the nontangential and the radialmaximal functions. Further, we may dominate M F, gmax by the flag maximal operator (see Section2.5), so M F, gmax is L p -bounded for all p ∈ (1 , ∞ ].We have seen in Sections 4 and 3 that H F, area ( H ν ) ⊆ H F, atom ( H ν ) ⊆ H F, gmax ( H ν ). The remarksabove show that H F, gmax ( H ν ) ⊆ H F, nontan ( H ν ) ⊆ H F, radial ( H ν ). We complete the identification ofthese five spaces by showing that H F, nontan ( H ν ) ⊆ H F, area ( H ν ) and H F, radial ( H ν ) ⊆ H F, nontan ( H ν ).Here, all inclusions are continuous, and there are also corresponding norm inequalities.63 .1 The inclusion H F, nontan ( H ν ) ⊆ H F, area ( H ν ) Let p (1) r denote the Poisson kernel associated to ∆ (1) on H ν . It is easy to see that, if f ∈ L p ( H ν ), then u , given by u ( r, g ) := f ∗ (1) p r ( g ) for all ( r, g ) ∈ R + × H ν , is harmonic, in the sense that ∆ (1) u = 0,where ∆ (1) = ∆ (1) − ∂ r (recall that the operator ∆ (1) was normalised to be positive). Moreover, if u is a harmonic function on R + × H ν , that is, ∆ (1) u = 0, then∆ (1) ( u )( r, g ) = 2 (cid:12)(cid:12)(cid:12) ∇ (1) u ( r, g ) (cid:12)(cid:12)(cid:12) (6.4)in R + × H ν , where ∇ (1) = ( ∇ (1) , ∂ r ) = ( X , . . . , X ν , Y , . . . , Y ν , ∂ r ) . (6.5)Write p (2) s for the usual Poisson kernel on R , and ∇ (2) for the full gradient on R × R +, that is, ∇ (2) = (cid:0) ∂ x , ∂ s (cid:1) ∀ ( x, s ) ∈ R × R + , and we define ∆ (2) := ∂ x − ∂ s . As usual, we shall write p r,s for p (1) r ∗ (2) p (2) s . Definition 6.4.
For f ∈ L ( H ν ), the Lusin–Littlewood–Paley area integral S F, area , p ( f ) of f associatedto the Poisson kernel is defined by S F, area , p ( f )( g ) := (cid:18)Z Z R + × R + (cid:12)(cid:12) f ∗ (1) r ∇ (1) p (1) r ∗ (2) s ∇ (2) p (2) s (cid:12)(cid:12) ∗ (1) χ (1) r ∗ (2) χ (2) s ( g ) drr dss (cid:19) / . (6.6)where χ (1) r and χ (2) s are the normalised indicator functions of the unit balls in H ν and R , as in Section2.1.We may write the area integral S F, area , p ( f ) more briefly as (cid:18)Z Z R + × R + (cid:12)(cid:12) f ∗ (1) ( / ∇ p ) r,s (cid:12)(cid:12) ∗ (1) χ r,s ( g ) drr dss (cid:19) / where / ∇ p r,s denotes the tensor-valued function ( ∇ (1) p (1) ) r ∗ (2) ( ∇ (2) p (2) ) s and χ r,s = χ (1) r ∗ (2) χ (2) s .Now Z A ( α ) S F, area , p ( f ) ( g ) dg = Z A ( α ) Z Z R + × R + (cid:12)(cid:12) f ∗ (1) ( / ∇ p ) r,s (cid:12)(cid:12) ∗ (1) χ r,s ( g ) drr dss dg = Z A ( α ) Z H ν Z Z R + × R + (cid:12)(cid:12) f ∗ (1) ( / ∇ p ) r,s ( gg ′ ) (cid:12)(cid:12) χ r,s ( g ′− ) drr dss dg ′ dg ≤ Z H ν Z H ν Z Z R + × R + H( gg ′ , r, s ) (cid:12)(cid:12) f ∗ (1) ( / ∇ p ) r,s ( gg ′ ) (cid:12)(cid:12) χ r,s ( g ′− ) drr dss dg ′ dg = Z H ν Z H ν Z Z R + × R + H( gg ′ , r, s ) (cid:12)(cid:12) f ∗ (1) ( / ∇ p ) r,s ( gg ′ ) (cid:12)(cid:12) χ r,s ( g ′− ) drr dss dg dg ′ = Z H ν Z Z R + × R + H( g, r, s ) (cid:12)(cid:12) f ∗ (1) ( / ∇ p ) r,s ( g ) (cid:12)(cid:12) drr dss dg, (6.7)provided that H : H ν × R + × R + → [0 , ∞ ) is chosen such that H( gg ′ , r, s ) = 1 whenever g ∈ A ( α ) and g ′ ∈ supp( χ r,s ). Theorem 6.5.
Suppose that u ∗ ∈ L ( H ν ) . Then S F, area , p ( f ) ∈ L ( H ν ) and kS F, area , p ( f ) k L ( H ν ) . k u ∗ k L ( H ν ) . roof. We need to estimate the left hand side of this inequality, and the trick is to choose H suitably.First, we take α ∈ R + and f ∈ L ( H ν ) such that k u ∗ k L ( H ν ) < ∞ , and defineΩ( α ) := { g ∈ H ν : u ∗ ( g ) ≤ α } , A ( α ) := (cid:26) g ∈ H ν : M F ( Ω( α ) c )( g ) < (cid:27) W := [ g ∈ A ( α ) Γ( g ) and f W := [ g ∈ A ( α ) c Γ β ( g ) , for some sufficiently large β , as determined after (6.9) below. Here Γ( g ) and Γ β ( g ) are as defined in(1.1). By definition and the L -boundedness of the flag maximal function M F , we see thatΩ( α ) c ⊆ A ( α ) c and | A ( α ) c | . | Ω( α ) c | . Set V ( g, r, s ) := Ω( α ) ∗ (1) p r,s ( g ) , ∀ ( g, r, s ) ∈ H ν × R + × R + . Recall from Lemma 2.21 that p r,s & | T ( o, r, s ) | − T ( o,r,s ) ; it follows that V ( g ′ , r, s ) = Z Ω( α ) p r,s (( g ′′ ) − g ′ ) dg ′′ = Z Ω( α ) p r,s (( g ′ ) − g ′′ ) dg ′′ & | Ω( α ) ∩ T ( g ′ , r, s ) || T ( g ′ , r, s ) | . When ( g ′ , r, s ) ∈ S g ∈ A ( α ) Γ( g ), there exists g ∈ A ( α ) such that g ∈ T ( g ′ , r, s ), and so | Ω( α ) c ∩ T ( g ′ , r, s ) || T ( g ′ , r, s ) | ≤ M F ( Ω( α ) c )( g ) < , which implies that | Ω( α ) ∩ T ( g ′ , r, s ) || T ( g ′ , r, s ) | ≥ . Hence there is a geometric constant C P ∈ (0 ,
1) such that V ( g ′ , r, s ) ≥ C P ∀ ( g ′ , r, s ) ∈ W. (6.8)Further, we claim that there is a geometric constant C P, in (0 , C P ) such that V ( g ′ , r, s ) ≤ C P, ∀ ( g ′ , r, s ) ∈ f W . (6.9)Given that R H ν p r,s ( g ) dg = 1, this follows similarly once the constant β is chosen large enough.Now, we choose a smooth function η : R → [0 ,
1] such that η ( t ) = 1 when t ≥ C P and η ( t ) = 0when t ≤ C P, . From (6.8), η ( V ( g ′ , r, s )) = 1 when ( g ′ , r, s ) ∈ W , and so, from (6.7), Z A ( α ) S F, area , p ( f )( g ) dg ≤ Z H ν Z Z R + × R + (cid:12)(cid:12) f ∗ (1) ( / ∇ p ) r,s ( g ′ ) η ( V ( g ′ , r, s )) (cid:12)(cid:12) drr dss dg ′ = Z H ν Z Z R + × R + (cid:12)(cid:12)(cid:12) f ∗ (2) ∇ (2) p (2) s ∗ (1) ∇ (1) p r ( g ′ ) η ( V ( g ′ , r, s )) (cid:12)(cid:12)(cid:12) r dr s ds dg ′ . (6.10)Set U ( g ′ , r, s ) = f ∗ (2) ∇ (2) p (2) s ∗ (1) p (1) r ( g ′ ). From (6.4), the last integrand may be written as a sum: (cid:12)(cid:12)(cid:12) ∇ (1) U ( g ′ , r, s ) η ( V ( g ′ , r, s )) (cid:12)(cid:12)(cid:12) = 12 ∆ (1) (cid:0) U ( g ′ , r, s ) η ( V ( g ′ , r, s )) (cid:1) U ( g ′ , r, s ) ∇ (1) U ( g ′ , r, s ) η ( V ( g ′ , r, s )) η ′ ( V ( g ′ , r, s )) ∇ (1) V ( g ′ , r, s ) − U ( g ′ , r, s ) η ′ ( V ( g ′ , r, s )) (cid:12)(cid:12)(cid:12) ∇ (1) V ( g ′ , r, s ) (cid:12)(cid:12)(cid:12) − U ( g ′ , r, s ) η ( V ( g ′ , r, s )) η ′′ ( V ( g ′ , r, s )) (cid:12)(cid:12)(cid:12) ∇ (1) V ( g ′ , r, s ) (cid:12)(cid:12)(cid:12) =: a ( g ′ , r, s ) + a ( g ′ , r, s ) + a ( g ′ , r, s ) + a ( g ′ , r, s ) , say. Thus, the right-hand side of (6.10) is bounded by P j =1 A j , where, for each j ,A j = (cid:12)(cid:12)(cid:12)(cid:12)Z H ν Z Z R + × R + a j ( g ′ , r, s ) r dr s ds dg ′ (cid:12)(cid:12)(cid:12)(cid:12) . We simplify the estimation by assuming that our functions are real-valued; otherwise one considersthe real and imaginary parts separately.We treat the term A using integration by parts and the decay of the Poisson kernel:A = 12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R + (cid:20) ∇ (1) (cid:18) U ( g ′ , r, s ) η (cid:0) ( Ω( α ) ∗ (1) p r,s ( g ′ ) (cid:1) (cid:19) r (cid:12)(cid:12)(cid:12)(cid:12) | g ′ | = ∞ ,r =0 ,r = ∞ − Z R + Z R Z C ν ∂ r (cid:18) U ( g ′ , r, s ) η (cid:0) ( Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:1) (cid:19) dz ′ dt ′ dr (cid:21) s ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 12 Z R + Z R Z C ν U ( g ′ , r, s ) η (cid:0) ( Ω( α ) ∗ (1) p r,s ( g ′ ) (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) r =0 dg ′ s ds = 12 Z R + Z R Z C ν (cid:12)(cid:12) ∇ (2) F ∗ (2) p (2) s ( g ′ ) (cid:12)(cid:12) η (cid:0) ( Ω( α ) ∗ (2) p (2) s ( g ′ ) (cid:1) dg ′ s ds For the term A , H¨older’s inequality yieldsA ≤ (cid:18)Z R Z R + Z C ν Z R + (cid:12)(cid:12) ∇ (1) U (( z ′ , t ′ ) , r, s ) (cid:12)(cid:12) η (cid:0) ( Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:1) dz ′ r dr dt s ds (cid:19) / × (cid:18)Z R Z R + Z C ν Z R + (cid:12)(cid:12) U (( z ′ , t ′ ) , r, s ) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) η ′ (cid:0) ( Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:1) ∇ (1) Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:12)(cid:12)(cid:12) dz ′ r dr dt ′ s ds (cid:19) / ≤ Z R Z R + Z C ν Z R + (cid:12)(cid:12) ∇ (1) U (( z ′ , t ′ ) , r, s ) (cid:12)(cid:12) η (cid:0) ( Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:1) dz ′ r dr dt ′ s ds + C Z R Z R + Z C ν Z R + (cid:12)(cid:12) U (( z ′ , t ′ ) , r, s ) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) η ′ (cid:0) ( Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:1) ∇ (1) Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:12)(cid:12)(cid:12) dz ′ r dr dt ′ s ds =: A + A . It is easy to see that A may be absorbed by the right-hand side of (6.10), and that A is dominatedby A . To treat A , we take a smooth function Φ on R such thatΦ ′ ( t ) = (cid:0) ( η ′ ( t )) + ( η ( t ) η ′ ( t )) (cid:1) / . It is easy to see that Φ ′ ( t ) is supported in [ C P, , C P ]. Then A may be bounded in much the sameway as A and A . Consequently, the right-hand side of (6.10) is bounded by C Z R + Z R Z C ν (cid:12)(cid:12) ∇ (2) F ∗ (2) p (2) s ( z ′ , t ′ ) (cid:12)(cid:12) η (cid:0) ( Ω( α ) ∗ (2) p (2) s ( z ′ , t ′ ) (cid:1) dz ′ dt ′ s ds C Z R Z R + Z C ν Z R + (cid:12)(cid:12) U (( z ′ , t ′ ) , r, s ) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) Φ ′ (cid:0) ( Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:1) ∇ (1) Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:12)(cid:12)(cid:12) dz ′ r dr dt ′ s ds =: B + B . For B , by repeating the estimate above for the right-hand side of (6.10),B = C Z C ν Z R + Z R (cid:12)(cid:12) ∇ (2) F ∗ (2) p (2) s ( z ′ , t ′ ) (cid:12)(cid:12) η (cid:0) ( Ω( α ) ∗ (2) p (2) s ( z ′ , t ′ ) (cid:1) dt ′ s ds dz ′ ≤ C Z C ν Z R (cid:12)(cid:12) F ( z ′ , t ′ ) (cid:12)(cid:12) η (cid:0) ( Ω( α ) ( z ′ , t ′ ) (cid:1) dt ′ dz ′ + C Z C ν Z R + Z R (cid:12)(cid:12) F ∗ (2) p (2) s ( z ′ , t ′ ) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) η ′ (cid:0) ( Ω( α ) ∗ (2) p (2) s ( z ′ , t ′ ) (cid:1)(cid:12)(cid:12)(cid:12) × (cid:12)(cid:12)(cid:12) ∇ (2) Ω( α ) ∗ (2) p (2) s ( z ′ , t ′ ) (cid:12)(cid:12)(cid:12) dz ′ dt ′ s ds + C Z C ν Z R + Z R (cid:12)(cid:12) F ∗ (2) p (2) s ( z ′ , t ′ ) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) η (cid:0) ( Ω( α ) ∗ (2) p (2) s ( z ′ , t ′ ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) η ′′ (cid:0) ( Ω( α ) ∗ (2) p (2) s ( z ′ , t ′ ) (cid:1)(cid:12)(cid:12)(cid:12) × (cid:12)(cid:12)(cid:12) ∇ (2) Ω( α ) ∗ (2) p (2) s ( z ′ , t ′ ) (cid:12)(cid:12)(cid:12) dz ′ dt ′ s ds. The definition of the function u ∗ implies that | F ( z ′ , t ′ ) | ≤ u ∗ ( z ′ , t ′ ) and (cid:12)(cid:12) F ∗ (2) p (2) s ( z ′ , t ′ ) (cid:12)(cid:12) ≤ u ∗ ( z ′ , t ′ ).Moreover, from the definition of the functions Ω( α ) and η , we see that u ∗ ( z ′ , t ′ ) ≤ α for all ( z ′ , t ′ ) and s such that η (cid:0) ( Ω( α ) ∗ (2) p (2) s ( z ′ , t ′ ) (cid:1) = 0 and η ′ (cid:0) ( Ω( α ) ∗ (2) p (2) s ( z ′ , t ′ ) (cid:1) = 0. HenceB ≤ C Z { ( z ′ ,t ′ ): u ∗ ( z ′ ,t ′ ) ≤ α } (cid:12)(cid:12) u ∗ ( z ′ , t ′ ) (cid:12)(cid:12) dt ′ dz ′ + Cα Z C ν Z R + Z R (cid:12)(cid:12)(cid:12) s ∇ Ω( α ) ∗ (2) p (2) s ( z ′ , t ′ ) (cid:12)(cid:12)(cid:12) dz ′ dt ′ dss . Note also that s ∇ (2) ∗ (2) p (2) s = 0, and so Z R + Z R (cid:12)(cid:12)(cid:12) s ∇ (2) Ω( α ) ∗ (2) p (2) s ( z ′ , t ′ ) (cid:12)(cid:12)(cid:12) dt ′ dss = Z R Z R + (cid:12)(cid:12)(cid:12) s ∇ (2) (1 − Ω( α ) ) ∗ (2) p (2) s ( z ′ , t ′ ) (cid:12)(cid:12)(cid:12) dss dt ′ ≤ C k − Ω( α ) ( z ′ , t ′ ) k L ( R ) . As a consequence, we deduce thatB ≤ C Z { ( z ′ ,t ′ ): u ∗ ( z ′ ,t ′ ) ≤ α } (cid:12)(cid:12) u ∗ ( z ′ , t ′ ) (cid:12)(cid:12) dt ′ dz ′ + Cα (cid:12)(cid:12)(cid:8) ( z ′ , t ′ ) : u ∗ ( z ′ , t ′ ) > α (cid:9)(cid:12)(cid:12) . We turn to the term B , which, up to a constant, is equal to Z R + Z C ν Z R + Z R (cid:12)(cid:12) ∇ (2) F ∗ (1) p (2) s ∗ (2) p (1) r ( z ′ , t ′ ) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) Φ (cid:0) ( Ω( α ) ∗ (1) ∇ (1) p r,s ( z ′ , t ′ ) (cid:1)(cid:12)(cid:12)(cid:12) dt ′ s ds dz ′ r dr. We now point out that (cid:12)(cid:12) ∇ (2) F ∗ (1) p (2) s ∗ (2) p (1) r ( z ′ , t ′ ) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ∇ (1) Φ (cid:0) ( Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:1)(cid:12)(cid:12)(cid:12) = 12 ∆ (2) (cid:18) u (( z ′ , t ′ ) , r, s ) ∇ (1) Φ (cid:0) ( Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:1) (cid:19) − u (( z ′ , t ′ ) , r, s ) ∇ (2) u (( z ′ , t ′ ) , r, s ) ∇ (1) Φ (cid:0) ( Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:1) ∇ (2) ∇ (1) Φ (cid:0) ( Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:1) − U (( z ′ , t ′ ) , r, s ) (cid:12)(cid:12)(cid:12) ∇ (2) ∇ (1) Φ (cid:0) ( Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:1)(cid:12)(cid:12)(cid:12) U (( z ′ , t ′ ) , r, s ) ∇ (1) Φ (cid:0) ( Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:1) ∇ (2) ∇ (2) ∇ (1) Φ (cid:0) ( Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:1) =: b + b + b + b , say. Then we see that B ≤ B + B + B + B , whereB i = (cid:12)(cid:12)(cid:12)(cid:12) C Z R + Z C ν Z R + Z R b i dt ′ s ds dz ′ r dr (cid:12)(cid:12)(cid:12)(cid:12) when i = 1 , , ,
4, For the term B , integration by parts shows thatB ≤ Z R + Z C ν Z R | F ∗ (1) p (1) r ( z ′ , t ′ ) | (cid:12)(cid:12)(cid:12) ∇ (1) Φ (cid:0) ( Ω( α ) ∗ (1) p (1) r ( z ′ , t ′ ) (cid:1)(cid:12)(cid:12)(cid:12) dt ′ dz ′ r dr ≤ α Z C ν Z R Z R + (cid:12)(cid:12)(cid:12) r ∇ (1) Ω( α ) ∗ (1) p (1) r ( z ′ , t ′ ) (cid:12)(cid:12)(cid:12) drr dt ′ dz ′ ≤ α Z C ν Z R Z R + (cid:12)(cid:12)(cid:12) (1 − Ω( α ) ) ∗ (1) r ∇ (1) p (1) r ( z ′ , t ′ ) (cid:12)(cid:12)(cid:12) drr dt ′ dz ′ ≤ Cα k − Ω( α ) ( z ′ , t ′ ) k L ( R ) = Cα (cid:12)(cid:12)(cid:8) ( z ′ , t ′ ) : u ∗ ( z ′ , t ′ ) > α (cid:9)(cid:12)(cid:12) , where the second inequality follow from the identity ∇ (1) Φ (cid:0) Ω( α ) ∗ (1) p (1) r ( z ′ , t ′ ) (cid:1) = Φ ′ (cid:0) Ω( α ) ∗ (1) p (1) r ( z ′ , t ′ ) (cid:1) ∇ (1) Ω( α ) ∗ (1) p (1) r ( z ′ , t ′ ) , and the estimates (cid:12)(cid:12) Φ ′ (cid:0) Ω( α ) ∗ (1) p (1) r ( z ′ , t ′ ) (cid:1)(cid:12)(cid:12) ≤ C and | F ∗ (1) p (1) r ( z ′ , t ′ ) | ≤ u ∗ ( z ′ , t ′ ) ≤ α , which followsfrom the support condition on Φ.For the term B , using H¨older’s inequality and then the same argument as for A , we see thatB may be dominated by1100 Z R + Z C ν Z R + Z R (cid:12)(cid:12) ∇ (2) F ∗ (1) p r,s ( z ′ , t ′ ) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ∇ (1) Φ (cid:0) ( Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:1)(cid:12)(cid:12)(cid:12) dt ′ s ds dz ′ r dr + Z R + Z C ν Z R + Z R (cid:12)(cid:12) F ∗ (1) p r,s ( z ′ , t ′ ) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ∇ (2) ∇ (1) Φ (cid:0) ( Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:1)(cid:12)(cid:12)(cid:12) dt ′ s ds dz ′ r dr =: B + B . We see immediately that B may be absorbed by B . To estimate B , we note that ∇ (2) ∇ (1) Φ (cid:0) ( Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:1) = Φ ′′ (cid:0) ( Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:1) ∇ (2) Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) ∇ (1) Ω( α ) ∗ (1) p r,s ( z ′ , t ′ )+ Φ ′ (cid:0) ( Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:1) ∇ (2) ∇ (1) Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) . Hence, B is further bounded by B + B with the above two integrands respectively. By usingthe support condition on Φ ′′ we see that B is dominated by α Z R + Z C ν Z R + Z R (cid:12)(cid:12)(cid:12) ∇ (2) Ω( α ) ∗ (1) p (2) s ∗ (2) p (1) r ( z ′ , t ′ ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ∇ (1) Ω( α ) ∗ (2) p (1) r ∗ (1) p (2) s ( z ′ , t ′ ) (cid:12)(cid:12)(cid:12) dt ′ s ds dz ′ r dr . α Z R + Z C ν Z R + Z R (cid:12)(cid:12)(cid:12) M H ν (cid:12)(cid:12) s ∇ (2) Ω( α ) ∗ (2) p (2) s (cid:12)(cid:12) ( z ′ , t ′ ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) M R (cid:12)(cid:12) r ∇ (1) Ω( α ) ∗ (1) p (1) r (cid:12)(cid:12) ( z ′ , t ′ ) (cid:12)(cid:12)(cid:12) dt ′ dss dz ′ drr , M H ν and M R are the Hardy–Littlewood maximal functions on H ν and R . Then by H¨older’sinequality,B ≤ Cα Z C ν Z R Z R + (cid:12)(cid:12)(cid:12) M H ν (cid:12)(cid:12) s ∇ (2) Ω( α ) ∗ (2) p (2) s (cid:12)(cid:12) ( z ′ , t ′ ) (cid:12)(cid:12)(cid:12) dss × Z R + (cid:12)(cid:12)(cid:12) M R (cid:12)(cid:12) r ∇ (1) Ω( α ) ∗ (1) p (1) r (cid:12)(cid:12) ( z ′ , t ′ ) (cid:12)(cid:12)(cid:12) drr dt ′ dz ′ ≤ Cα ( Z C ν Z R (cid:18)Z R + (cid:12)(cid:12)(cid:12) M H ν (cid:12)(cid:12) s ∇ (2) Ω( α ) ∗ (2) p (2) s (cid:12)(cid:12) ( z ′ , t ′ ) (cid:12)(cid:12)(cid:12) dss (cid:19) dt ′ dz ′ ) / × ( Z C ν Z R (cid:18)Z R + (cid:12)(cid:12)(cid:12) M R (cid:12)(cid:12) r ∇ (1) Ω( α ) ∗ (1) p (1) r (cid:12)(cid:12) ( z ′ , t ′ ) (cid:12)(cid:12)(cid:12) drr (cid:19) dt ′ dz ′ ) / ≤ Cα ( Z C ν Z R (cid:18)Z R + (cid:12)(cid:12)(cid:12) s ∇ (2) (1 − Ω( α ) ) ∗ (2) p (2) s ( z ′ , t ′ ) (cid:12)(cid:12)(cid:12) dss (cid:19) dt ′ dz ′ ) / × ( Z C ν Z R (cid:18)Z R + (cid:12)(cid:12)(cid:12) (1 − Ω( α ) ) ∗ (1) r ∇ (1) p (1) r ( z ′ , t ′ ) (cid:12)(cid:12)(cid:12) drr (cid:19) dt ′ dz ′ ) / ≤ Cα k − Ω( α ) k L ( H ν ) · k − Ω( α ) k L ( H ν ) = Cα (cid:12)(cid:12)(cid:8) ( z ′ , t ′ ) : u ∗ ( z ′ , t ′ ) > α (cid:9)(cid:12)(cid:12) , where the third inequality follows from the Fefferman–Stein vector-valued inequality for the Hardy–Littlewood maximal function, and from replacement of Ω( α ) by 1 − Ω( α ) (which is possible since both s ∇ (2) p (2) s and r ∇ (1) p (1) r have cancellation); the fourth inequality follows from Littlewood–Paley theory.Next, by the support condition on Φ ′ and the cancellation properties of s ∇ (2) p (2) s and r ∇ (1) p (1) r we seethat B ≤ Cα Z R + Z C ν Z R + Z R (cid:12)(cid:12)(cid:12) Ω( α ) ∗ (1) ∇ (1) p (1) r ∗ (2) ∇ (2) p (2) s ( z ′ , t ′ ) (cid:12)(cid:12)(cid:12) dt ′ s ds dz ′ r dr = Cα Z R + Z C ν Z R + Z R (cid:12)(cid:12)(cid:12) (1 − Ω( α ) ) ∗ (1) r ∇ (1) p (1) r ∗ (2) s ∇ (2) p (2) s ( z ′ , t ′ ) (cid:12)(cid:12)(cid:12) dt ′ dss dz ′ drr ≤ Cα k − Ω( α ) k L ( H ν ) = Cα (cid:12)(cid:12)(cid:8) ( z ′ , t ′ ) : u ∗ ( z ′ , t ′ ) > α (cid:9)(cid:12)(cid:12) , where the second inequality follows from Littlewood–Paley theory.The term B may be handled in the same way as B .We now turn to B . Note that ∇ (1) Φ (cid:0) ( Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:1) ∇ (2) ∇ (2) ∇ (1) Φ (cid:0) ( Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:1) = Φ ′ (cid:0) ( Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:1) Φ ′′′ (cid:16) Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:17) × |∇ (1) Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) | |∇ (2) Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) | + 2Φ ′ (cid:0) ( Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:1) Φ ′′ (cid:0) ( Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:1) ∇ (1) Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) × ∇ (2) Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) ∇ (1) ∇ (2) Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) . Hence, B is bounded by B +B , where these terms involve the two terms above in the integrands.The term B may be handled in the same way as B . H¨older’s inequality implies that B may69e dominated by a multiple of α Z R + Z C ν Z R + Z R (cid:12)(cid:12)(cid:12) ∇ (2) Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ∇ (1) Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:12)(cid:12)(cid:12) dt ′ s ds dz ′ r dr + α Z R + Z C ν Z R + Z R (cid:12)(cid:12)(cid:12) ∇ (1) Ω( α ) ∗ (1) p r,s ( z ′ , t ′ ) (cid:12)(cid:12)(cid:12) dt ′ s ds dz ′ r dr, which is bounded by Cα (cid:12)(cid:12)(cid:8) ( z ′ , t ′ ) : u ∗ ( z ′ , t ′ ) > α (cid:9)(cid:12)(cid:12) .Combining all the estimates above we see that Z A ( α ) S F, area , p ( f )( g ′ ) dg ′ . α | Ω( α ) c | + Z Ω( α ) (cid:12)(cid:12) u ∗ ( g ′ ) (cid:12)(cid:12) dg ′ . Hence (cid:12)(cid:12) Ω( α ) c (cid:12)(cid:12) ≤ | Ω( α ) c ∩ A ( α ) c | + | Ω( α ) c ∩ A ( α ) | . | Ω( α ) c | + 1 α Z A ( α ) S F, area , p ( f )( g ′ ) dg ′ . | Ω( α ) c | + 1 α Z Ω( α ) (cid:12)(cid:12) u ∗ ( g ′ ) (cid:12)(cid:12) dg ′ . This implies that kS F, area , p ( u ) k L ( H ν ) ≤ C k u ∗ k L ( H ν ) . H F, radial ( H ν ) ⊆ H F, nontan ( H ν ) In this section, we show that the radial maximal function Hardy space is contained in the nontangentialmaximal function Hardy space. Once this is done, it follows that all the maximal function Hardy spacescoincide with the atomic Hardy space.
Theorem 6.6.
Suppose that ( ϕ (1) , ϕ (2) ) and ( φ (1) , φ (2) ) are Poisson-bounded pairs, as in Definition2.20, and that R H ν ϕ (1) ( g ) dg = R R ϕ (2) ( t ) dt = R H ν φ (1) ( g ) dg = R R φ (2) ( t ) dt = 1 . Then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup ( g,r,s ) ∈ Γ( · ) (cid:12)(cid:12) f ∗ (1) ϕ r,s ( g ) (cid:12)(cid:12)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( H ν ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup r,s ∈ R + (cid:12)(cid:12) f ∗ (1) φ r,s ( · ) (cid:12)(cid:12)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( H ν ) ∀ f ∈ L ( H ν ) , (6.11) where ϕ r,s = ϕ (1) r ∗ (2) ϕ (2) s and φ r,s is defined similarly.Proof. It suffices to prove the following estimate: if ( ψ (1) , ψ (2) ) is a Poisson-bounded pair, as inDefinition 2.20, and R H ν ψ (1) = R R ψ (2) = 0, then for every f ∈ L ( H ν ),sup ( g,r,s ) ∈ Γ( g ) | f ∗ (1) ψ r,s ( g ) | . (cid:26) M (cid:16) M (cid:0) sup r,s ∈ R + | f ∗ (1) φ r,s ( g ) | θ (cid:1)(cid:17)(cid:27) /θ (6.12)for all θ in (0 ,
1) that are sufficiently close to 1.To prove (6.12), we apply the discrete version of Calder´on’s reproducing formula. The properties of φ (1) and φ (2) mean that we can treat them as approximations to the identity. We define the differenceoperator τ j,k = τ (1) j ∗ (2) τ (2) k , where τ (1) j := φ (1) j − φ (1) j +1 and τ (2) k := φ (2) k − φ (2) k +1 . Then by repeating theproof of Lemma 2.25 with minor modifications, we may establish the reproducing formula: f ( g ) = X j,k ∈ Z X R ∈ R ( N,j,k ) | R | e τ j,k ( g, g R ) f ∗ (1) τ j,k ( g R ) (6.13)70here the points g R = ( z R , t R ) are arbitrary in R and the equality holds in L p ( H ν ) and in the flag testfunction space as in [32], and R ( N, j, k ) is the set of adapted rectangles such that width( R ) = 2 − j − N and height( R ) = 2 − k − N , where N is a large positive integer to be determined later, and e τ j,k ( g, cent( R )),as a function of g , satisfies the same size and regularity estimates and has the same cancellationproperties as τ j,k ( g ).Substituting this reproducing formula into the convolution in left-hand side of (6.12), we see that | f ∗ (1) ψ r,s ( g ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:28) X j,k ∈ Z X R ∈ R ( N,j,k ) | R | e τ j,k ( · , g R ) ∗ (1) ψ r,s ( g ) f ∗ (1) τ j,k ( g R ) (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) . We recall the almost orthogonality estimate of [32, Lemma 6]: for g = ( z , t ), g = ( z, t ), and r and s such that ( g, r, s ) ∈ Γ( g ), | e τ j,k ( · , g R ) ∗ (1) ψ r,s ( g ) | . (cid:16) − j r ∧ r − j (cid:17) β (cid:16) − k s ∧ s − k (cid:17) β (2 − j ∨ r ) γ (2 − j ∨ r + | z − z R | ) ν +2 γ (2 − k ∨ s ) γ (2 − k ∨ s + | t − t R | ) γ if (2 − j ∨ r ) ≤ − k ∨ s, (cid:16) − j r ∧ r − j (cid:17) β (cid:16) − k s ∧ s − k (cid:17) β (2 − j ∨ r ) γ (2 − j ∨ r + | z − z R | ) ν + γ (2 − j ∨ r ) γ (2 − j ∨ r + √ | t − t R | ) γ if (2 − j ∨ r ) ≥ − k ∨ s, . (cid:16) − j r ∧ r − j (cid:17) β (cid:16) − k s ∧ s − k (cid:17) β (2 − j ∨ r ) γ (2 − j ∨ r + | z − z R | ) ν +2 γ (2 − k ∨ s ) γ (2 − k ∨ s + | t − t R | ) γ if (2 − j ∨ r ) ≤ − k ∨ s, (cid:16) − j r ∧ r − j (cid:17) β (cid:16) − k s ∧ s − k (cid:17) β (2 − j ∨ r ) γ (2 − j ∨ r + | z − z R | ) ν + γ (2 − j ∨ r ) γ (2 − j ∨ r + √ | t − t R | ) γ if (2 − j ∨ r ) ≥ − k ∨ s, We consider only the case where (2 − j ∨ r ) ≤ − k ∨ s , as the other case may be treated similarly.In this case, | f ∗ (1) ψ r,s ( g ) | . X j,k (2 − j ∨ r ) ≤ − k ∨ s X R ∈ R ( N,j,k ) (cid:18) − j r ∧ r − j (cid:19) β (cid:18) − k s ∧ s − k (cid:19) β (2 − j ∨ r ) γ (2 − j ∨ r + | z − z R | ) ν +2 γ × (2 − k ∨ s ) γ (2 − k ∨ s + | t − t R | ) γ | R | inf ˜ g ∈ R (cid:12)(cid:12) f ∗ (1) τ j,k (˜ g ) (cid:12)(cid:12) . Note that (cid:12)(cid:12) f ∗ (1) τ j,k (˜ g ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) f ∗ (1) τ (1) j ∗ (2) τ (2) k (˜ g ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) f ∗ (1) (cid:0) φ (1) j − φ (1) j +1 (cid:1) ∗ (2) (cid:0) φ (2) k − φ (2) k +1 (cid:1) (˜ g ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ r,s ∈ R + (cid:12)(cid:12) f ∗ (1) φ r,s (˜ g ) (cid:12)(cid:12) . For every θ ∈ (0 , | f ∗ (1) ψ r,s ( g ) | . ( X j,k (2 − j ∨ r ) ≤ − k ∨ s X R ∈ R ( N,j,k ) (cid:18) − j r ∧ r − j (cid:19) βθ (cid:18) − k s ∧ s − k (cid:19) βθ (2 − j ∨ r ) γθ (2 − j ∨ r + | z − z R | ) (2 ν +2 γ ) θ × (2 − k ∨ s ) γθ (2 − k ∨ r + | t − t R | ) (1+ γ ) θ | R | θ inf ˜ g ∈ R sup r,s ∈ R + (cid:12)(cid:12) f ∗ (1) φ r,s (˜ g ) (cid:12)(cid:12) θ ) /θ . We split the sum P R ∈ R ( N,j,k ) into annuli based on the distance of R from the point g , and see that | f ∗ (1) ψ r,s ( g ) | X j,k (2 − j ∨ r ) ≤ − k ∨ s (cid:18) − j r ∧ r − j (cid:19) βθ (cid:18) − k s ∧ s − k (cid:19) βθ − N (2 ν +1) /r ′ × [( r − n ∧ nj )2 − nj ( s − ∧ k )2 − k ] /r ′ ( M F (cid:18) sup r,s ∈ R + (cid:12)(cid:12) f ∗ (1) φ r,s ( · ) (cid:12)(cid:12) θ (cid:19) ( g ) ) /θ . ( M F (cid:18) sup r,s ∈ R + (cid:12)(cid:12) f ∗ (1) φ r,s ( · ) (cid:12)(cid:12) θ (cid:19) ( g ) ) /θ , where the last inequality holds if (2 ν + 1) / (2 ν + 1 + β ) < θ < ψ (1) ( g ) = (cid:0) ϕ (1) ( g ) − φ (1) ( g − · g ) (cid:1) and ψ (2) ( t ) = (cid:0) ϕ (2) ( t ) − φ (2) ( t − t ) (cid:1) .Then when g = ( z , t ), g ∈ H ν , and r, s >
0, such that ( g, r, s ) ∈ Γ( g ), (cid:12)(cid:12) f ∗ (1) ϕ r,s ( g ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) f ∗ (1) ϕ (1) r ∗ (2) ϕ (2) s ( g ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) f ∗ (1) [2 ψ (1) r + φ (1) r ] ∗ (2) [2 ψ (2) s + φ (2) s ]( g ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) f ∗ (1) ψ (1) r ∗ (2) ψ (2) s ( g ) (cid:12)(cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12)(cid:12) f ∗ (1) ψ (1) r ∗ (2) φ (2) s ( g ) (cid:12)(cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12)(cid:12) f ∗ (1) φ (1) r ∗ (2) ψ (2) s ( g ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) f ∗ (1) φ (1) r ∗ (2) φ (2) s ( g ) (cid:12)(cid:12)(cid:12)(cid:12) =: A + A + A + A . Now A is bounded by n M F (cid:0) sup r,s ∈ R + (cid:12)(cid:12) f ∗ (1) φ r,s ( · ) (cid:12)(cid:12) θ (cid:1) ( g ) o /θ by the calculation above, and A isbounded by sup r,s ∈ R + (cid:12)(cid:12) f ∗ (1) φ r,s ( g ) (cid:12)(cid:12) by definition. To treat A and A , we iterate one-parameterestimates, and see that these too are controlled by n M F (cid:0) sup r,s ∈ R + (cid:12)(cid:12) f ∗ (1) φ r,s ( · ) (cid:12)(cid:12) θ (cid:1) ( g ) o /θ .The proof of Theorem 6.6 is complete. In this section, we complete the proof of Theorem 1.1 by showing that flag Riesz transforms alsocharacterise the flag Hardy space, and the putting everything together to conclude the proof. H F, Riesz ( H ν ) ⊆ H F, dis ( H ν ) In this section, we show that the Hardy space defined by singular integrals is a subspace of the squarefunction Hardy space. We remind the reader of a definition.
Definition 7.1.
The (tensor-valued) flag Riesz transformation is given by R F = ∇ (1) ∆ − / ∇ (2) ∆ − / ,and the space H F, Riesz ( H ν ) is the set of all f ∈ L ( H ν ) such that ∇ (1) ∆ − / ( f ), ∇ (2) ∆ − / ( f ) and R F ( f ) all lie in L ( H ν ), with norm k f k H F, Riesz ( H ν ) := k f k L ( H ν ) + (cid:13)(cid:13)(cid:13) ∇ (1) ∆ − / ( f ) (cid:13)(cid:13)(cid:13) L ( H ν ) + (cid:13)(cid:13)(cid:13) ∇ (2) ∆ − / ( f ) (cid:13)(cid:13)(cid:13) L ( H ν ) + kR F ( f ) k L ( H ν ) . (7.1)We are going to prove a slightly more general result, for which we need some more notation.Let {R j : j = 0 , . . . , J } be a collection of simple singular integral operators on H ν that characterise72he Folland–Stein–Christ–Geller Hardy space H F SCG ( H ν ); that is, f ∈ H F SCG ( H ν ) if and only if R j f ∈ L ( H ν ) for each j . In particular, the identity operator together with the Riesz transformswould do. In this case, k f k H F SCG ( H ν ) ≃ J X j =0 kR j f k L ( H ν ) , by the closed graph theorem. For more information, see [8]. We write H for the Hilbert transform inthe central variable on H ν , that is, H f ( z, t ) = Z R f ( z, t − s ) dss . Definition 7.2.
The space H F, sing ( H ν ) is the set of all functions f ∈ L ( H ν ) for which R j f ∈ L ( H ν )and HR j f ∈ L ( H ν ) when j = 0 , . . . , J , with norm k f k H F, sing ( H ν ) := J X j =0 kR j f k L ( H ν ) + k H R j f k L ( H ν ) . (7.2) Theorem 7.3.
Let R j and H be as above. If f ∈ L ( H ν ) and R j f ∈ L ( H ν ) and HR j f ∈ L ( H ν ) ,where j = 0 , . . . , J , then f ∈ H F, dis ( H ν ) and k f k H F, dis ( H ν ) . k f k H F, sing ( H ν ) .Proof. We combine the characterisation of Christ and Geller with a randomisation argument. Beforewe begin our proof we remind the reader that the operators R j and H commute.We recall from the classical theory of Hardy spaces that f ∈ H ( R ) if and only if both f ∈ L ( R )and H f ∈ L ( R ), and that the H¨ormander–Mihlin multiplier theorem on R shows that if m is abounded function on R , differentiable except perhaps at 0, such that y ym ′ ( y ) is also bounded,then the associated Fourier multiplier operator f
7→ F − ( m ˆ f ) is bounded from H ( R ) to L ( R ) (here F indicates the Fourier transformation f ˆ f on R ). For the details, see, for instance, [9]. A similarresult holds for the one-parameter Hardy space on H ν , except that more differentiability of m isrequired; it suffices that the functions y y k m ( k ) ( y ) are bounded for k between 0 and n + 2, but morederivatives may be used without any changes other than in the value of some unimportant constants.Let η be a smooth function on R + , supported in [1 / , P n ∈ Z η (2 n y ) = 1 for all y ∈ R + . Then by spectral theory, there are smooth functions ϕ (1 ,m ) on H ν and ϕ (2 ,n ) on R such that f ∗ (1) ϕ (1 ,m ) = η (2 m ∆ (1) ) f and f ∗ (2) ϕ (2 ,n ) = η (2 n ∆ (2) ) f for all f ∈ S ( H ν ).In this paragraph, we consider functions on R . Let s n : Ω → C be a collection of independentRademacher random variables (that is, each takes the values ± ω ∈ Ω , the operator A ω from H ( R ) to L ( R ), defined (at least formally) by A ω f = X n ∈ Z s n ( ω ) f ∗ (2) ϕ (2 ,n ) , is bounded, with norm bounded independently of ω , by the Mikhlin–H¨ormander multiplier theorem.That is, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X n ∈ Z s n ( ω ) f ∗ (2) ϕ (2 ,n ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( R ) . k f k L + kH f k L ( R ) for all f ∈ H ( R ), or equivalently, for all f ∈ L ( R ) such that H f ∈ L ( R ).73e take a function f on H ν such that R j f and HR j f lie in L ( H ν ), and apply the precedingobservation to the functions R j f ( z, · ) and HR j f ( z, · ). We then integrate over C ν and deduce that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) R j X n ∈ Z s n ( ω ) f ∗ (2) ϕ (2 ,n ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L . kR j f k L + kHR j f k L . Now P n ∈ Z s n ( ω ) f ∗ (2) ϕ (2 ,n ) ∈ H F SCG ( H ν ) from the Folland–Stein–Christ–Geller characterisation [8,Theorem A], and so, similarly, if r m : Ω → C is another family of independent Rademacher randomvariables, independent of the first family, then X m ∈ Z X n ∈ Z r m ( ω ) s n ( ω ) f ∗ (2) ϕ (2 ,n ) ∗ (1) ϕ (1 ,m ) = X m ∈ Z r m ( ω ) X n ∈ Z s n ( ω ) f ∗ (2) ϕ (2 ,n ) ∗ (1) ϕ (1 ,m ) and this function lies in L ( H ν ) and Z H ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X m ∈ Z r m ( ω ) X n ∈ Z s n ( ω ) f ∗ (2) ϕ (2 ,n ) ∗ (1) ϕ (1 ,m ) ( g ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dg . J X j =1 (cid:16) kR j f k L ( H ν ) + kHR j f k L ( H ν ) (cid:17) . It now follows from the orthonormality of the functions r m and s n , Khinchin’s inequality, Minkowski’sinequality, and Khinchin’s inequality that X m ∈ Z X n ∈ Z (cid:12)(cid:12)(cid:12) f ∗ (2) ϕ (2 ,n ) ∗ (1) ϕ (1 ,m ) ( g ) (cid:12)(cid:12)(cid:12) ! / = Z Ω Z Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X m ∈ Z X n ∈ Z r m ( ω ) s n ( ω ) f ∗ (2) ϕ (2 ,n ) ∗ (1) ϕ (1 ,m ) ( g ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dω dω / . Z Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X m ∈ Z X n ∈ Z r m ( ω ) s n ( ω ) f ∗ (2) ϕ (2 ,n ) ∗ (1) ϕ (1 ,m ) ( g ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dω / ≤ Z Ω Z Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X m ∈ Z X n ∈ Z r m ( ω ) s n ( ω ) f ∗ (2) ϕ (2 ,n ) ∗ (1) ϕ (1 ,m ) ( g ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dω / dω . Z Ω Z Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X m ∈ Z X n ∈ Z r m ( ω ) s n ( ω ) f ∗ (2) ϕ (2 ,n ) ∗ (1) ϕ (1 ,m ) ( g ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dω dω . By integrating over H ν and using Fubini’s theorem and the estimates above, we conclude that Z H ν X m ∈ Z X n ∈ Z (cid:12)(cid:12)(cid:12) f ∗ (2) ϕ (2 ,n ) ∗ (1) ϕ (1 ,m ) ( g ) (cid:12)(cid:12)(cid:12) ! / dg . Z Z Ω × Ω Z H ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X m ∈ Z r m ( ω ) X n ∈ Z s n ( ω ) f ∗ (2) ϕ (2 ,n ) ∗ (1) ϕ (1 ,m ) ( g ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dg dω dω ≤ Z Z Ω × Ω J X j =0 (cid:16) kR j f k L ( H ν ) + kHR j f k L (cid:17) dω dω = k f k H F, Riesz ( H ν ) , as required.The reader who is uncomfortable with our formal calculations may take sums over finite subsets of Z in the arguments above, and then allow these subsets to become arbitrarily large. It may also be worthpointing out that we could replace f ∗ (2) ϕ (2 ,n ) ∗ (1) ϕ (1 ,m ) in the proof above by f ∗ (1) ϕ (1 ,m ) ∗ (2) ϕ (2 ,n ) .74 .2 Conclusion of the proof of Theorem 1.1 and remarks We have already shown in Section 3 that various singular integrals, including the Riesz transforms,map H F, atom ( H ν ) into L ( H ν ), and in Section 5 that H F, dis ( H ν ) = H F, atom ( H ν ) (once again, all theseinclusions are continuous, and corresponding norm inequalities hold). Theorem 7.3 therefore completesour characterisation of the Hardy space by singular integrals, and completes our proof of Theorem1.1.We are now able to sharpen the results about certain singular integral operators. Corollary 7.4.
The Hilbert transformation H and simple singular integral operators, as defined inSection 2.6, are bounded on H F ( H ν ) .Proof. Take a family of simple singular integral operators R j that characterise the Folland–Stein–Christ–Geller space H F SCG ( H ν ), and f in H F ( H ν ). Then R j f and R j H f all lie in H F SCG ( H ν ). As H is the identity, R j ( H f ) and R j H ( H f ) all lie in H F SCG ( H ν ), so that H f ∈ H F ( H ν ).Similarly, if K is a simple singular integral operator, and f ∈ H F ( H ν ), then R j K f ∈ L ( H ν )since the composition R j K is a linear combination of the identity and another simple singular integraloperator. Further, commutativity and the same argument shows that R j HK f = R j KH f ∈ L ( H ν )since H f ∈ H F ( H ν ). We therefore conclude that K f ∈ H F ( H ν ). Remark . On the one hand, the main theorem tells us that H F ( H ν ) can be characterised by simplesingular integrals. On the other hand, the definition of an atom involves cancellations, in the sensethat we impose conditions of the form a = ∆ M (1) ∆ N (2) b for some function b , and these imply that R H ν a ( g ) p ( g ) dg = 0 for certain polynomials of low degree. It is now apparent that, provided that M > ν/ N ≥
1, we end up with the same atomic Hardy space.Similarly, the other versions of the Hardy space involve auxiliary functions (usually written ϕ (1) and ϕ (2) ). It is now also clear that the Hardy space is largely independent of these functions, as longas they satisfy the relevant decay and cancellation conditions. The main results in this section are an isomorphism of our Hardy space with the flag Hardy spaceof Han, Lu and Sawyer [32], and the consequent identification of the interpolation space between anyof our Hardy spaces and L ( H ν ), and proofs that certain flag singular integrals are bounded on ourspace H F ( H ν ) to L ( H ν ), The last main result is on sharp Marcinkiewicz multiplier, Proposition 1.6. H F ( H ν ) via heat semigroups In the proofs of Theorems 3.11 and 4.7, we did not need to use much about the functions ϕ (1) and ϕ (2) . However, we did use translation and dilation arguments to reduce estimates involving particlesto estimates involving particles centred at the origin and of a certain size. This was for convenienceand to simplify the geometry.We assert that our methods also work in settings that are not translation-invariant. Hence, if wehave self-adjoint operators L and L whose associated heat semigroups have kernels with Gaussian75pper bounds, and analogous estimates for derivatives, and have the conservation property e − s L e − t L s L and L by the formula S F, L , L ( f )( z, u ) := (cid:18)Z R Z ∞ Z C n Z ∞ χ r,s (( z, u ) · ( z ′ , u ′ ) − ) × (cid:12)(cid:12) ( r L e − r L s L e − s L ) f ( z ′ , u ′ ) (cid:12)(cid:12) dz ′ drr n +3 du ′ dss (cid:19) / , and the flag Hardy space H F, L , L ( H ν ) associated to L and L to be the completion under the norm kS F, L , L ( f ) k L ( H ν ) of the space { f ∈ L ( H ν ) : S F, L , L ( f ) ∈ L ( H ν ) } . Our arguments (appropriatelyextended) show that the Hardy space H F, L , L ( H ν ) coincides with our H F ( H ν ).This continues the theory of Duong and McIntosh [15], Duong and Yan [16, 17], Hofmann andMayboroda [35] (and others) on singular integrals with nonsmooth kernels and function spaces asso-ciated to operators, and we expect that our Hardy space may be applied to the study of more typesof differential equations on H ν . H HLS ( H ν ) By definition, the reflection map R : f f (( · ) − ) is a linear bijection of our flag Hardy space H F ( H ν ) and the Han–Lu–Sawyer [32] space H HLS ( H ν ). This correspondence has some interestingconsequences. Proposition 8.1.
The complex interpolation spaces between our flag Hardy space H F ( H ν ) and theLebesgue space L ( H ν ) are the Lebesgue spaces L p ( H ν ) where < p < .Proof. The map R from our flag Hardy space H F ( H ν ) to the Han–Lu–Sawyer [32] space H HLS ( H ν )is an isometry of all the Lebesgue spaces L p ( H ν ). Our interpolation theorem is now an immediatecorollary of theirs. Corollary 8.2.
Suppose that < p < ∞ . Then kS F, area , ϕ ( f ) k L p ( H ν ) . p k f k L p ( H ν ) ∀ f ∈ L p ( H ν ) . Proof.
From Theorem 1.1 and Proposition 4.6, S F, area , ϕ is bounded from H F, area ( H ν ) to L ( H ν ) andfrom L ( H ν ) to L ( H ν ), and hence by interpolation S F, area , ϕ is bounded on L p ( H ν ) when 1 < p < L p ( H ν ) when 2 < p < ∞ . Corollary 8.3.
Suppose that ( ϕ (1) , ϕ (2) ) is a Poisson-bounded pair, as in Definition 5.1. Then thesublinear maps S F, cts , ϕ and S F, dis , ϕ are bounded on L p ( H ν ) when < p < ∞ .Proof. This also follows from Theorem 1.1, interpolation and duality. H F ( H ν ) is a proper subspace of the Hardy space H F SCG ( H ν ) Proof of Proposition 1.3.
The singular integral characterisation of our Hardy space H F ( H ν ) showsimmediately that it is a subspace of the space H F SCG ( H ν ).To see that it is a proper subspace, we consider the function a , defined by a ( z, t ) = ψ ( z ) ϕ ( t ) , ∀ ( z, t ) ∈ H ν , where 76. ψ in C ∞ ( C ν ) is supported in the unit ball of C ν , Z C ν ψ ( z ) dz = 0, and Z C ν | ψ ( z ) | dz = 1; and2. ϕ in C ∞ ( R ) is supported in ( − , Z R ϕ ( t ) dt = 1.It is easy to see that a is a multiple of an atom of the Folland–Stein–Christ–Geller Hardy space H F SCG ( H ν ), but that the L ( H ν ) norm of the radial maximal function a ∗ is infinite, and hence a / ∈ H F ( H ν ). In this section, we prove Proposition 1.4 and 1.5.
Proof of Proposition 1.4.
Let K be the convolution operator given by K f = f ∗ K , with the kernel K as in (1.3). By Proposition 3.10 and Theorem 1.1, it suffices to show that there exist C, δ , δ ∈ R + such that Z ( S ∗ ) c |K ( a R )( g ′ ) | dg ′ ≤ Cρ δ ,δ ( R, S ) | R | / k a R k (8.1)for all particles a R associated to an arbitrary adapted rectangle R and all adapted rectangles S thatcontain R . By translation and dilation invariance, we may suppose that R ⊆ R ∗ = T ( o, , h ), so that | R | h | R ∗ | h h , and further that S ∗ = T ( o, r ∗ , h ∗ ), where r ∗ ≥ h ∗ ≥ h .To continue, note that ( S ∗ ) c ⊂ I ∪ II ∪ III ∪ IV, whereI := { ( z, u ) ∈ H ν : | z | ≤ ν, | u | > h ∗ } II := { ( z, u ) ∈ H ν : | z | > ν, | u | > h ∗ }} III := { ( z, u ) ∈ H ν : | z | > r ∗ , | u | > νh } IV := { ( z, u ) ∈ H ν : | z | > r ∗ , | u | ≤ νh } . From (1.3), it is easy to verify that | ∂ αz ∂ βu K ( z, u ) = | ∂ αz ∂ βu K ( z, u ) | . | z | − ν − α ( | z | + | u | / ) − − β ∀ ( z, u ) ∈ H ν . Much as in Section 3.4, from this regularity estimate and the properties of particles, we see that Z II ∪ III |K ( a R )( g ′ ) | dg ′ ≤ Cρ δ ,δ ( R, S ) | R | / k a R k for small δ , δ ∈ R + .To estimate the term R I |K ( a R )( g ′ ) | dg ′ , it suffices to control the norm of the operator ˜ K on L ( C ν )given by ˜ K ( f ) = f ∗ (0) ˜ K with convolution kernel ˜ K , where the parameter u satisfies | u | > h ∗ , and ∗ (0) denotes the standard convolution on C ν . The operator norm can be estimated from the L ∞ normof the Fourier transform of ˜ K , which may be computed as follows: b ˜ K ( ξ , u ) = Z C ν ∂ βu (cid:16) ω ( z )( | z | + u ) ν | z | + iu (cid:17) e − iz · ξ dz ∀ ξ ∈ C ν . Since ω ( z ) is smooth and homogeneous of degree 0, via polar coordinates, it suffices to note thatsup ξ (cid:12)(cid:12) b ˜ K ( ξ , u ) (cid:12)(cid:12) . | u | β ∀| u | > h ∗ , with arbitrary z ′ on the unit sphere of C ν . Hence the integral of |K ( a R )( g ′ ) | over I is bounded by Cρ ,β ( R, S ) | R | / k a R k . The estimate of R IV |K ( a R )( g ′ ) | dg ′ is simpler. Hence (8.1) holds, and the proof is complete.77 roof of Proposition 1.5. This proof is very similar to the previous proof, and we leave the reader tofill in the details.
Remark . It is worth noting that the Phong–Stein singular integral operator is a prototypical flagsingular integral: the kernel is not homogeneous, but a product of homogeneous factors. The lasttwo proofs extend to cover more general flag singular operators, and show that these map H F ( H ν ) to L ( H ν ). The fact that flag singular integral operators form an algebra, proved by Nagel, Ricci, Steinand Wainger [51, 52], then implies that flag singular operators map H F ( H ν ) to itself, by an argumentlike that of the proof of Corollary 7.4. m (∆ (1) , i T ) In this section, we prove Proposition 1.6.We begin by reducing matters to a slightly simpler case. If m satisfies the condition of Proposition1.6, so does ˇ m , defined by ˇ m ( µ, λ ) = m ( µ, − λ ). It follows that it suffices to consider separately thecases where m is even and odd in the second variable. The case where m ( µ, λ ) = sgn( λ ) correspondsto the Hilbert transform in the central variable, which is bounded on H F ( H ν ) by Corollary 7.4. Thusit suffices to consider the case where m is even.Our sharp Marcinkiewicz multiplier theorem will be proved using weighted L estimates, which goback to work of M¨uller, Ricci and Stein [47], and we now introduce the relevant weight functions. Definition 8.5.
For positive ε , we define the weight function w εj,ℓ : H ν → R + by w εj,ℓ ( z, t ) := 2 − ν ( j + ℓ ) (1 + 2 j + ℓ | z | ) ν (1+ ε ) − ℓ (1 + 2 ℓ | t | ) ε ∀ ( z, t ) ∈ H ν . For convenience, we also set w ε ,j,ℓ ( g ) := 2 − ν ( j + ℓ ) (1 + 2 j + ℓ | z | ) ν (1+ ε ) and w ε ,j,ℓ ( g ) := 2 − ℓ (1 + 2 ℓ | t | ) ε . Before we restate our theorem, we extend a result of M¨uller, Ricci and Stein [47] very mildly.
Lemma 8.6.
Suppose that Ψ (1) and Ψ (2) are smooth, [0 , -valued functions on R + , supported in aninterval (1 /c, c ) , for some c ∈ R + , and extend Ψ (2) to a smooth even function on R , still denoted Ψ (2) .Let K j,ℓ be the convolution kernel of the operator m j,ℓ (∆ (1) / |T | , i T ) := m (∆ (1) / |T | , i T ) Ψ (1) (2 − j ∆ (1) / |T | ) Ψ (2) (2 − ℓ i T ) . (8.2) Then (cid:18)Z H ν | K j,ℓ ( g ) | w εj,ℓ ( g ) dg (cid:19) / . c,ε (cid:13)(cid:13)(cid:13) η , m j , ℓ (cid:13)(cid:13)(cid:13) L α,β , (8.3) when α = ν + ε and β = (1 + ε ) / .Proof. If µ ∈ R + and λ ∈ R , and Ψ (1) (2 − j µ/ | λ | ) Ψ (2) (2 − ℓ λ ) = 0, then 2 − ℓ | λ | ∈ (1 /c, c ) and 2 − j µ/ | λ | ∈ (1 /c, c ); it follows that µ ∈ (2 j + ℓ /c, j + ℓ c ).Inequality (8.3), in the case where c is close to 1, is proved in Proposition 5.3 and Lemma 2.5 of[47]. To remove the restriction on c , we observe that when c is large, the kernel in question may bewritten as a finite sum of rescaled versions of kernels with support in (1 /c ′ , c ′ ), where c ′ is close to 1,apply the result of [47] to these kernels, and then sum.78rom the Gaussian upper bounds of the heat kernels of ∆ (1) and ∆ (2) , we have the followingoff-diagonal estimates. Lemma 8.7.
Let η be a C ∞ c function with supp η ⊂ [1 / , . E, F are closed sets of H ν and E R , F R areclosed sets of R . d K and d R are the Kor´anyi distance on H ν and Euclidean distance on R respectively.Then for any s ≥ k η (2 − j ∆ (1) ) k L ( E ) → L ( F ) ≤ C s (2 j/ d K ( E, F )) − s , (8.4) k η (2 − j i T ) k L ( E R ) → L ( F R ) ≤ C s (2 j d R ( E R , F R )) − s . (8.5) Proof.
The proof is standard, and we sketch it briefly. From the the Gaussian upper bounds of heatkernels of ∆ (1) , for all s > | K η (2 − j ∆ (1) ) ( g, g ′ ) | ≤ C s V ( g, j/ )(1 + 2 j/ d K ( g, g ′ )) − s . Then for functions f, f ′ ∈ L ( H ν ) with supp f ⊂ E and supp f ′ ⊂ F , we have (cid:12)(cid:12)(cid:12)(cid:12)Z H ν η (2 − j ∆ (2) )( f )( g ) f ′ ( g ) dg (cid:12)(cid:12)(cid:12)(cid:12) ≤ C s Z F Z E V ( g, j/ )(1 + 2 j/ d K ( g, g ′ )) − s | f ( g ) || f ′ ( g ) | dg dg ′ ≤ C s (2 j/ d K ( E, F )) s Z F Z E V ( g, j/ )(1 + 2 j/ d K ( g, g ′ )) − s | f ( g ) || f ′ ( g ) | dg dg ′ ≤ C s (2 j/ d K ( E, F )) s (cid:18)Z F Z E V ( g, j/ )(1 + 2 j/ d K ( g, g ′ )) − s | f ( g ) | dg dg ′ (cid:19) / (cid:18)Z F Z E V ( g, j/ )(1 + 2 j/ d K ( g, g ′ )) − s | f ′ ( g ) | dg dg ′ (cid:19) / ≤ C s (2 j/ d K ( E, F )) − s k f k k f ′ k . Hence (8.4) follows readily. The proof of (8.5) is similar.We are now ready to prove Proposition 1.6.
Proof of Proposition 1.6.
We take a smooth, even, [0 , η on R , with support in theset ( − , − / ∪ (1 / , P j ∈ Z (cid:12)(cid:12) η (2 − j µ ) (cid:12)(cid:12) = 1 for all µ ∈ R \ { } . By Proposition 3.10 andTheorem 1.1, it suffices to show that there exist C, δ , δ ∈ R + such that Z ( S ∗ ) c (cid:18)X j,ℓ (cid:12)(cid:12)(cid:12) m (∆ (1) / |T | , i T ) η j,ℓ (∆ (1) , i T )( a R )( g ′ ) (cid:12)(cid:12)(cid:12) (cid:19) / dg ′ ≤ Cρ δ ,δ ( R, S ) | R | / k a R k (8.6)for all particles a R associated to R ∈ R and all S ∈ R that contain R , where η j,ℓ (∆ (1) , i T ) := η (2 − j − ℓ ∆ (1) ) η (2 − ℓ i T ). By translation and dilation invariance, we may suppose that R ⊆ R ∗ = T ( o, , h ) , so | R | h | R ∗ | h h , and also that S ∗ = T ( o, r ∗ , h ∗ ) , where r ∗ ≥ h ∗ ≥ h .We take a smooth, compactly supported, nonnegative-real-valued function Ψ (2) on R + that takesthe value 1 on (1 / ,
2) and vanishes off (1 / , (2) to a smooth even function on R , stilldenoted Ψ (2) ; we also take a smooth, compactly supported, nonnegative-real-valued function Ψ (1) on R + that takes the value 1 on (1 / ,
4) and vanishes off (1 / , (1) (2 − j µ/ | λ | ) Ψ (2) (2 − ℓ λ ) η (2 − j − ℓ µ ) η (2 − ℓ λ ) = η (2 − j − ℓ µ ) η (2 − ℓ λ )79nd hence m ( µ/ | λ | , λ ) Ψ (1) (2 − j µ/ | λ | ) Ψ (2) (2 − ℓ λ ) η (2 − j − ℓ µ ) η (2 − ℓ λ ) = m ( µ/ | λ | , λ ) η (2 − j − ℓ µ ) η (2 − ℓ λ )for all µ ∈ R + and all λ ∈ R . Thus to prove (8.6), it suffices to show that Z ( S ∗ ) c (cid:18)X j,ℓ (cid:12)(cid:12)(cid:12) m j,ℓ (∆ (1) / |T | , i T ) η j,ℓ (∆ (1) , i T )( a R )( g ′ ) (cid:12)(cid:12)(cid:12) (cid:19) / dg ′ . ρ δ ,δ ( R, S ) | R | / k a R k . (8.7)We define four regions in H ν using the Euclidean metric:I := { ( z, u ) ∈ H ν : | z | ≤ , | u | > h ∗ } II := { ( z, u ) ∈ H ν : | z | > , | u | > max { h ∗ , ν ( h + | z | ) }} III := { ( z, u ) ∈ H ν : | z | > r ∗ , | u | > ν ( h + | z | ) } IV := { ( z, u ) ∈ H ν : | z | > r ∗ , | u | ≤ ν ( h + | z | ) } . Clearly I ∪ II ∪ III ∪ IV ∪ S ∗ = H ν . Now we estimate the integral on the left-hand side of (8.7) on theregions II ∪ III, I and IV.
First part:
We estimate the integral on the left-hand side of (8.7) on II ∪ III. Since ℓ ⊆ ℓ , Z II ∪ III (cid:18)X j,ℓ (cid:12)(cid:12)(cid:12) m j,ℓ (∆ (1) / |T | , i T ) η j,ℓ (∆ (1) , i T )( a R )( g ′ ) (cid:12)(cid:12)(cid:12) (cid:19) / dg ′ ≤ X j,ℓ Z II ∪ III (cid:12)(cid:12)(cid:12) m j,ℓ (∆ (1) / |T | , i T ) η j,ℓ (∆ (1) , i T )( a R )( g ′ ) (cid:12)(cid:12)(cid:12) dg ′ ≤ X j,ℓ Z R Z II ∪ III (cid:12)(cid:12)(cid:12) K m j,ℓ (∆ (1) / |T | ,i T ) η j,ℓ (∆ (1) ,i T ) ( g − g ′ ) (cid:12)(cid:12)(cid:12) dg ′ (cid:12)(cid:12) a R ( g ) (cid:12)(cid:12) dg. (8.8)Change variable g − g ′ = g ′′ and note that g ′ = ( z ′ , u ′ ) ∈ II and g = ( z, u ) ∈ R imply that | u ′ − u − S ( z ′ , z ) | = | u ′ − u − ν Im z ′ ¯ z | ≥ | u ′ − u | / g ′′ = g − g ′ ∈ II ∗ = { ( z ′′ , u ′′ ) : | z ′′ | > / , | u ′′ | > h ∗ / } . It follows from Lemma 8.6 that Z II ∗ (cid:12)(cid:12)(cid:12) K m j,ℓ (∆ (1) / |T | ,i T ) η j,ℓ (∆ (1) ,i T ) ( g ′′ ) (cid:12)(cid:12)(cid:12) dg ′′ ≤ Z II ∗ (cid:12)(cid:12)(cid:12) K m j,ℓ (∆ (1) / |T | ,i T ) η j,ℓ (∆ (1) ,i T ) ( g ′′ ) (cid:12)(cid:12)(cid:12) w εj,ℓ ( g ′′ ) dg ′′ Z II ∗ w εj,ℓ ( g ′′ ) dg ′′ ≤ C Z | z | > / Z | u | >h ∗ / ν ( j + ℓ ) (1 + 2 j + ℓ | z | ) − ν (1+ ε ) ℓ (1 + 2 ℓ | u | ) − − ε dzdu ≤ C − ( j + ℓ ) ε/ (2 ℓ h ∗ ) − ε = C − ( j + ℓ ) ε/ (2 ℓ h ) − ε ( hh ∗ ) ε . (8.9)It follows similarly that Z III (cid:12)(cid:12)(cid:12) K m j,ℓ (∆ (1) / |T | ,i T ) η j,ℓ (∆ (1) ,i T ) ( g − g ′ ) (cid:12)(cid:12)(cid:12) dg ′ ≤ C (2 ( j + ℓ ) / ) − ε (2 ℓ h ) − ε ( 1 r ∗ ) ε . (8.10)80n the other hand, let η ′ ( λ ) := λ M η ( λ ) and η ′′ ( λ ) := λ N η ( λ ). Replacing η by η ′ or η ′′ in (8.8) for2 j + ℓ ≤ ℓ h ≤ Z II ∪ III (cid:12)(cid:12)(cid:12) m j,ℓ (∆ (1) / |T | , i T ) η j,ℓ (∆ (1) , i T )( a R )( g ′ ) (cid:12)(cid:12)(cid:12) dg ′ = Z II ∪ III (cid:12)(cid:12)(cid:12) m j,ℓ (∆ (1) / |T | , i T )(2 j + ℓ ) M (2 ℓ h ) N η ′ j + ℓ (∆ (1) ) η ′′ ℓ ( i T )( h − N ∆ − M (1) ∆ − N (2) a R )( g ′ ) (cid:12)(cid:12)(cid:12) dg ′ ≤ (2 j + ℓ ) M (2 ℓ h ) N Z R Z II ∪ III (cid:12)(cid:12)(cid:12) K m j,ℓ (∆ (1) / |T | ,i T ) η ′ j + ℓ (∆ (1) ) η ′′ ℓ ( i T ) ( g − g ′ ) (cid:12)(cid:12)(cid:12) dg ′ (cid:12)(cid:12) h − N b R ( g ) (cid:12)(cid:12) dg. (8.11)Note that η ′ and η ′′ satisfy the same conditions as η and we only use the L norm and support of thefunction width( R ) − M height( R ) − N b R and note that it has the same support as a R and its L normis bounded by that of a R . For convenience, we denote both width( R ) − M height( R ) − N b R and a R by a R and we denote both η ′ or η ′′ by η for 2 j + ℓ ≤ ℓ h ≤ X j,ℓ min { , j + ℓ } M min { , ℓ } N − ( j + ℓ ) ε/ (2 ℓ h ) − ε (( hh ∗ ) ε + ( 1 r ∗ ) ε ) | R | / k a R k L ≤ C (cid:18)(cid:18) hh ∗ (cid:19) ε + (cid:18) r ∗ (cid:19) ε (cid:19) | R | / k a R k L . (8.12) Second Part:
We divide I into dyadic pieces:I = [ k ≥ I u,k := [ k ≥ { ( z, u ) : | z | ≤ , k − h ∗ < | u | ≤ k h ∗ } . Then the integral of the second part is controlled by Z I (cid:18)X j,ℓ (cid:12)(cid:12)(cid:12) m j,ℓ (∆ (1) / |T | , i T ) η j,ℓ (∆ (1) , i T )( a R )( g ′ ) (cid:12)(cid:12)(cid:12) (cid:19) / dg ′ ≤ X k ≥ Z I u,k (cid:18)X j,ℓ (cid:12)(cid:12)(cid:12) m j,ℓ (∆ (1) / |T | , i T ) η ℓ ( i T ) η j + ℓ (∆ (1) )( a R )( g ′ ) (cid:12)(cid:12)(cid:12) (cid:19) / dg ′ . (8.13)Define sets A u,k := { ( z, u ) : z ∈ C ν , k − h ∗ − ν | z | < | u | ≤ k +1 h ∗ + 8 ν | z |} and B u,k := { ( z, u ) : z ∈ C ν , | u | > k +1 h ∗ + 8 ν | z | or | u | ≤ k − h ∗ − ν | z |} . Then H ν may decomposed into three pieces in different ways: H ν = A u,k ∪ ( B u,k \ R ∗ ) ∪ R ∗ . Note that the Sobolev norms of the multipler functions are the same and thus we regard the operator81 j,ℓ (∆ (1) / |T | , i T ) η ℓ ( i T ) the same as m j,ℓ (∆ (1) / |T | , i T ). Then Z I u,k (cid:18)X j,ℓ (cid:12)(cid:12)(cid:12) m j,ℓ (∆ (1) / |T | , i T ) η ℓ ( i T ) η j + ℓ (∆ (1) )( a R )( g ′ ) (cid:12)(cid:12)(cid:12) (cid:19) / dg ′ ≤ Z I u,k (cid:18)X j,ℓ (cid:12)(cid:12)(cid:12) m j,ℓ (∆ (1) / |T | , i T ) χ B u,k \ R ∗ η j + ℓ (∆ (1) )( a R )( g ′ ) (cid:12)(cid:12)(cid:12) (cid:19) / dg ′ + Z I u,k (cid:18)X j,ℓ (cid:12)(cid:12)(cid:12) m j,ℓ (∆ (1) / |T | , i T ) χ A u,k η j + ℓ (∆ (1) )( a R )( g ′ ) (cid:12)(cid:12)(cid:12) (cid:19) / dg ′ + Z I u,k (cid:18)X j,ℓ (cid:12)(cid:12)(cid:12) m j,ℓ (∆ (1) / |T | , i T ) χ R ∗ η j + ℓ (∆ (1) )( a R )( g ′ ) (cid:12)(cid:12)(cid:12) (cid:19) / dg ′ =: I Bu,k + I
Au,k + I
Ru,k . (8.14)For the term I Bu,k , it follows from Lemma 8.6 thatI
Bu,k ≤ X j,ℓ Z I u,k (cid:12)(cid:12)(cid:12) m j,ℓ (∆ (1) / |T | , i T ) χ B u,k \ R ∗ η j + ℓ (∆ (1) )( a R )( g ′ ) (cid:12)(cid:12)(cid:12) dg ′ ≤ X j,ℓ Z B u,k \ R ∗ Z I u,k (cid:12)(cid:12)(cid:12) K m j,ℓ (∆ (1) / |T | ,i T ) ( g − g ′ ) (cid:12)(cid:12)(cid:12) dg ′ (cid:12)(cid:12) η j + ℓ (∆ (1) )( a R )( g ) (cid:12)(cid:12) dg ≤ X j,ℓ Z B u,k \ R ∗ Z I u,k w εj,ℓ ( g − g ′ ) dg ′ ! / (cid:12)(cid:12) η j + ℓ (∆ (1) )( a R )( g ) (cid:12)(cid:12) dg. (8.15)Change variable g − g ′ = g ′′ and note that if g ′ = ( z ′ , u ′ ) ∈ I u,k and g = ( z, u ) ∈ B u,k then | u ′ − u − S ( z ′ , z ) | ≥ | u ′ − u | / g ′′ = g − g ′ ∈ { ( z ′′ , u ′′ ) ∈ H ν : | u ′′ | > k − h ∗ } . Hence Z I u,k w εj,ℓ ( g − g ′ ) dg ′ ≤ Z C ν Z | u ′′ | > k − h ∗ ( j + ℓ ) ν ℓ (1 + 2 j + ℓ | z ′′ | ) ν (1+ ε ) (1 + 2 ℓ | u ′′ | ) ε dz ′′ du ′′ . (2 k ℓ h ∗ ) − ε . (8.16)It follows from Lemma 8.7 that Z B u,k \ R ∗ (cid:12)(cid:12) η j + ℓ (∆ (1) )( a R )( g ) (cid:12)(cid:12) dg ≤ C − ( j + ℓ ) ε | R | / k a R k L . (8.17)A similar argument as in (8.11) and (8.12), with η replaced by η ′ = λ M η ( λ ) or η ′′ = λ N η ( λ ) in (8.14)for 2 j + ℓ ≤ ℓ h ≤ Bu,k ≤ C X j,ℓ min { , j + ℓ } M min { , ℓ h } N (2 k ℓ h ∗ ) − ε/ − ( j + ℓ ) ε | R | / k a R k L ≤ C − kε/ (cid:18) hh ∗ (cid:19) ε/ | R | / k a R k L . (8.18)82or the term I Au,k , replacing η by η ′ = λ M η ( λ ) or η ′′ = λ N η ( λ ) in (8.14) for 2 j + ℓ ≤ ℓ h ≤ Au,k ≤ X j,ℓ Z I u,k (cid:12)(cid:12)(cid:12) m j,ℓ (∆ (1) / |T | , i T ) χ A u,k η j + ℓ (∆ (1) )( a R )( g ′ ) (cid:12)(cid:12)(cid:12) dg ′ / | I u,k | / ≤ X j,ℓ k m j,ℓ (∆ (1) / |T | , i T ) k L → L Z A u,k (cid:12)(cid:12)(cid:12) η j + ℓ (∆ (1) )( a R )( g ) (cid:12)(cid:12)(cid:12) dg / | I u,k | / ≤ X j,ℓ min { , j + ℓ } M min { , ℓ h } N Z A u,k (cid:12)(cid:12)(cid:12) η j + ℓ (∆ (1) )( a R )( g ) (cid:12)(cid:12)(cid:12) dg / | I u,k | / . Note that d K ( A u,k , R ) ≥ √ k − h ∗ . It follows from Lemma 8.7 that for some s > ε Z A u,k (cid:12)(cid:12)(cid:12) η j + ℓ (∆ (1) )( a R )( g ) (cid:12)(cid:12)(cid:12) dg ≤ C (2 j + ℓ k h ∗ ) − s | R |k a R k L . Note that j >
0. Combining the last two inequalities shows thatI
Au,k ≤ X j> X ℓ min { , ℓ h } N (2 j + ℓ k h ∗ ) − s | R |k a R k L / | I u,k | / ≤ C − kε/ (cid:18) hh ∗ (cid:19) ε/ | R | / k a R k L . (8.19)For the term I Ru,k , replacing η by η ′ = λ M η ( λ ) or η ′′ = λ N η ( λ ) in (8.14) for 2 j + ℓ ≤ ℓ h ≤ Ru,k ≤ X j,ℓ Z I u,k (cid:12)(cid:12)(cid:12) m j,ℓ (∆ (1) / |T | , i T ) χ R ∗ η j + ℓ (∆ (1) )( a R )( g ′ ) (cid:12)(cid:12)(cid:12) dg ′ / | I u,k | / ≤ X j,ℓ k m j,ℓ (∆ (1) / |T | , i T ) k L ( R ∗ ) → L (I u,k ) Z H ν (cid:12)(cid:12)(cid:12) η j + ℓ (∆ (1) )( a R )( g ) (cid:12)(cid:12)(cid:12) dg / | I u,k | / . (8.20)By duality we show the estimate for k m j,ℓ (∆ (1) / |T | , i T ) k L (I u,k ) → L ( R ∗ ) . From Lemma 8.6, k m j,ℓ (∆ (1) / |T | , i T ) χ I u,k f k L ( R ∗ ) ≤ Z R ∗ Z I u,k | K m j,ℓ (∆ (1) / |T | ,i T ) ( g − g ′ ) | w ε ,j,ℓ ( g − g ′ ) dg Z I u,k w ε ,j,ℓ ( g − g ′ ) | f ( g ) | dg dg ′ ≤ ℓ (2 ℓ k h ∗ ) − (1+ ε ) Z R ∗ Z I u,k | K m j,ℓ (∆ (1) / |T | ,i T ) ( g − g ′ ) | w ε ,j,ℓ ( g − g ′ ) w ε ,j,ℓ ( g − g ′ ) dg Z I u,k w ε ,j,ℓ ( g − g ′ ) | f ( g ) | dg dg ′ . ℓ (2 ℓ k h ∗ ) − (1+ ε ) k η , m j , ℓ k L α,β Z R ∗ Z I u,k | f ( g ) | w ε ,j,ℓ ( g − g ′ ) dg dg ′ . ℓ (2 ℓ k h ∗ ) − (1+ ε ) min { , ℓ h } N | R | Z I u,k | f ( g ) | dg. Ru,k ≤ (cid:18)X j,ℓ ℓ (2 ℓ k h ∗ ) − (1+ ε ) min { , ℓ h } N | R | Z H ν (cid:12)(cid:12)(cid:12) η j + ℓ (∆ (1) )( a R )( g ) (cid:12)(cid:12)(cid:12) dg (cid:19) / | k h ∗ | / ≤ X ℓ ℓ (2 ℓ k h ∗ ) − (1+ ε ) min { , ℓ h } N | R | Z H ν (cid:12)(cid:12)(cid:12) a R ( g ) (cid:12)(cid:12)(cid:12) dg ! / | k h ∗ | / ≤ C − kε/ (cid:18) hh ∗ (cid:19) ε/ | R | / k a R k L . (8.21)Combining estimates (8.13), (8.14), (8.18) (8.19) and (8.21), Z I (cid:18)X j,ℓ (cid:12)(cid:12)(cid:12) m j,ℓ (∆ (1) / |T | , i T ) η j,ℓ (∆ (1) , i T )( a R )( g ′ ) (cid:12)(cid:12)(cid:12) (cid:19) / dg ′ ≤ C (cid:18) hh ∗ (cid:19) ε/ | R | / k a R k L . (8.22) Third Part:
We decompose IV into dyadic piecesIV = [ k ≥ IV z,k := [ k ≥ { ( z, u ) : 2 k − r ∗ < | z | ≤ k r ∗ , | u | ≤ ν ( h + | z | ) } . Without loss of generality, we can assume that 2 k r ∗ ≫ h , otherwise it is much easier. Note thatsupp η ℓ ( i T )( a R ) ⊂ { ( z, u ) : | z | < , u ∈ R } =: A R . Also note that the Sobolev norms of the multiplerfunctions are the same and thus we regard the operator m j,ℓ (∆ (1) / |T | , i T ) η j + ℓ (∆ (1) ) the same as m j,ℓ (∆ (1) / |T | , i T ). Then the integral of the third part is controlled by Z IV (cid:18)X j,ℓ (cid:12)(cid:12)(cid:12) m j,ℓ (∆ (1) / |T | , i T ) η j,ℓ (∆ (1) , i T )( a R )( g ′ ) (cid:12)(cid:12)(cid:12) (cid:19) / dg ′ ≤ X k ≥ Z IV z,k (cid:18)X j,ℓ (cid:12)(cid:12)(cid:12) m j,ℓ (∆ (1) / |T | , i T ) η ℓ ( i T )( a R )( g ′ ) (cid:12)(cid:12)(cid:12) (cid:19) / dg ′ ≤ X k ≥ Z IV z,k (cid:18)X j,ℓ (cid:12)(cid:12)(cid:12) m j,ℓ (∆ (1) / |T | , i T ) χ A R /R ∗ η ℓ ( i T )( a R )( g ′ ) (cid:12)(cid:12)(cid:12) (cid:19) / dg ′ + X k ≥ Z IV z,k (cid:18)X j,ℓ (cid:12)(cid:12)(cid:12) m j,ℓ (∆ (1) / |T | , i T ) χ R ∗ η ℓ ( i T )( a R )( g ′ ) (cid:12)(cid:12)(cid:12) (cid:19) / dg ′ =: IV + IV . (8.23)For the term IV , it follows from Lemma 8.6 thatIV ≤ X k ≥ X j,ℓ Z IV z,k (cid:12)(cid:12)(cid:12) m j,ℓ (∆ (1) / |T | , i T ) χ A R /R ∗ η ℓ ( i T )( a R )( g ′ ) (cid:12)(cid:12)(cid:12) dg ′ ≤ X k ≥ X j,ℓ Z A R /R ∗ Z IV z,k (cid:12)(cid:12)(cid:12) K m j,ℓ (∆ (1) / |T | ,i T ) ( g − g ′ ) (cid:12)(cid:12)(cid:12) dg ′ (cid:12)(cid:12) η ℓ ( i T )( a R )( g ) (cid:12)(cid:12) dg ≤ X k ≥ X j,ℓ min { , j + ℓ } M min { , ℓ h } N Z A R /R ∗ Z IV z,k w εj,ℓ ( g − g ′ ) dg ′ ! / (cid:12)(cid:12) η ℓ ( i T )( a R )( g ) (cid:12)(cid:12) dg ≤ X k ≥ X j,ℓ min { , j + ℓ } M min { , ℓ h } N (2 ( j + ℓ ) / k r ∗ ) − ε Z A R /R ∗ (cid:12)(cid:12) η ℓ ( i T )( a R )( g ) (cid:12)(cid:12) dg. (8.24)84t follows from Lemma 8.7 that Z A R /R ∗ (cid:12)(cid:12) η ℓ ( i T )( a R )( g ) (cid:12)(cid:12) dg ≤ C (2 ℓ h ) − ε | R | / k a R k L . (8.25)Combining (8.24) and (8.25) shows thatIV ≤ X k ≥ X j,ℓ min { , j + ℓ } M min { , ℓ h } N (2 ( j + ℓ ) / k r ∗ ) − ε (2 ℓ h ) − ε | R | / k a R k L ≤ C ( 1 r ∗ ) ε | R | / k a R k L . (8.26)Next we estimate the term IV . For a fixed g ′ = ( z ′ , u ′ ) ∈ IV z,k , we divide R to two sets A g ′ and B g ′ such that A g ′ := { g = ( z, u ) ∈ R : | u ′ − u − S ( z ′ , z ) | ≤ h } , B g ′ := { g = ( z, u ) ∈ R : | u ′ − u − S ( z ′ , z ) | > h } . Also define the projection sets of A g ′ by A g ′ ,z := { z : ( z, u ) ∈ A g ′ } , A g ′ ,u := { u : ( z, u ) ∈ A g ′ } . Weclaim that | A g ′ ,z | ≤ C ν h k r ∗ . (8.27)Next we prove the above claim. Let u ′ = 8 νt k r ∗ with a fixed 0 ≤ t ≤
1. Since the rotation does notchange the measure of sets on C ν , without loss of generality, we can assume that z ′ = ( | z ′ | , , . . . ,
0) =(2 s k r ∗ , , . . . ,
0) with a fixed 1 / ≤ s ≤
1. Then if z = ( x j + iy j ) ∈ A g ′ ,z , then | u ′ − u − S ( z ′ , z ) | = | νt k r ∗ − u − νs k r ∗ y | ≤ h implies that | y − ts | ≤ sν h k r ∗ . Note that g ∈ R implies that | z | ≤ C ν = 16 ν . FurtherIV ≤ X k ≥ Z IV z,k (cid:18)X j,ℓ (cid:12)(cid:12)(cid:12)Z A g ′ K m j,ℓ (∆ (1) / |T | ,i T ) ( g − g ′ ) η ℓ ( i T )( a R )( g ) dg (cid:12)(cid:12)(cid:12) (cid:19) / dg ′ + X k ≥ Z IV z,k (cid:18)X j,ℓ (cid:12)(cid:12)(cid:12)Z B g ′ K m j,ℓ (∆ (1) / |T | ,i T ) ( g − g ′ ) η ℓ ( i T )( a R )( g ) dg (cid:12)(cid:12)(cid:12) (cid:19) / dg ′ =: IV + IV . (8.28)For the term IV , it follows from Lemma 8.6 and similar argument as in (8.11) and (8.12), replacing η by η ′ = λ M η ( λ ) or η ′′ = λ N η ( λ ) for 2 j + ℓ ≤ ℓ h ≤ ≤ X k ≥ X j,ℓ Z IV z,k Z B g ′ (cid:12)(cid:12)(cid:12) K m j,ℓ (∆ (1) / |T | ,i T ) ( g − g ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) η ℓ ( i T ) a R ( g ) (cid:12)(cid:12) dg dg ′ ≤ X k ≥ X j,ℓ Z R Z { g ′ ∈ IV z,k : | u ′ − u − S ( z ′ ,z ) | > h } (cid:12)(cid:12)(cid:12) K m j,ℓ (∆ (1) / |T | ,i T ) ( g − g ′ ) (cid:12)(cid:12)(cid:12) dg ′ (cid:12)(cid:12) η ℓ ( i T ) a R ( g ) (cid:12)(cid:12) dg ≤ X k ≥ X j,ℓ min { , j + ℓ } M min { , ℓ h } N Z R Z { g ′ ∈ IV z,k : | u ′ − u − S ( z ′ ,z ) | > h } w εj,ℓ ( g − g ′ ) dg ′ ! / · (cid:12)(cid:12) η ℓ ( i T )( a R )( g ) (cid:12)(cid:12) dg ≤ X k ≥ X j,ℓ min { , j + ℓ } M min { , ℓ h } N (2 ( j + k ) / k r ∗ ) − ε (2 ℓ h ) − ε | R | / k a R k L ≤ C ( 1 r ∗ ) ε | R | / k a R k L . (8.29)85or the term IV , by the Cauchy-Schwarz inequality, we haveIV ≤ X k ≥ X j,ℓ Z IV z,k (cid:12)(cid:12)(cid:12)Z A g ′ K m j,ℓ (∆ (1) / |T | ,i T ) ( g − g ′ ) η ℓ ( i T )( a R )( g ) dg (cid:12)(cid:12)(cid:12) dg ′ / | IV z,k | / . (8.30)Using the Cauchy–Schwarz inequality, we see that Z IV z,k (cid:12)(cid:12)(cid:12)Z A g ′ K m j,ℓ (∆ (1) / |T | ,i T ) ( g − g ′ ) η ℓ ( i T )( a R )( g ) dg (cid:12)(cid:12)(cid:12) dg ′ ≤ Z IV z,k Z A g ′ ,z (cid:12)(cid:12)(cid:12)Z A g ′ ,u K m j,ℓ (∆ (1) / |T | ,i T ) ( g − g ′ ) η ℓ ( i T )( a R )( g ) du (cid:12)(cid:12)(cid:12) dz | A g ′ ,z | dg ′ ≤ C h k r ∗ Z IV z,k Z A g ′ ,z Z A g ′ ,u | K m j,ℓ (∆ (1) / |T | , − i T ) ( g − g ′ ) | w ε ,j,ℓ ( g − g ′ ) du ! Z A g ′ ,u w ε ,j,ℓ ( g − g ′ ) | η ℓ ( i T )( a R )( g ) | du ! dz dz ′ du ′ ≤ C h k r ∗ Z k − r ∗ < | z ′ |≤ k Z R Z | z |≤ (cid:18)Z R | K m j,ℓ (∆ (1) / |T | ,i T ) ( g − g ′ ) | w ε ,j,ℓ ( g − g ′ ) du (cid:19)(cid:18)Z R ℓ (1 + 2 ℓ | u ′ − u − S ( z ′ , z ) | ) ε | η ℓ ( i T )( a R )( g ) | du (cid:19) dz dz ′ du ′ ≤ C ( j + ℓ ) ν h k r ∗ (2 ( j + ℓ ) / k r ∗ ) − ν (1+ ε ) Z k − r ∗ < | z ′ |≤ k Z R Z | z |≤ (cid:18)Z R | K m j,ℓ (∆ (1) / |T | ,i T ) ( g − g ′ ) | w εj,ℓ ( g − g ′ ) du (cid:19)(cid:18)Z R ℓ (1 + 2 ℓ | u ′ − u − S ( z ′ , z ) | ) ε | η ℓ ( i T )( a R )( g ) | du (cid:19) dz dz ′ du ′ . (8.31)It follows from Lemma 8.6 that Z C ν Z R | K m j,ℓ (∆ (1) / |T | ,i T ) ( g − g ′ ) | w εj,ℓ ( g − g ′ ) dudz ′ = Z C ν Z R | K m j,ℓ (∆ (1) / |T | ,i T ) ( z ′ , u ′ , z, u ) | w εj,ℓ ( z ′ , u ′ , z, u ) dudz ′ = Z C ν Z R | K m j,ℓ (∆ (1) / |T | ,i T ) ( z ′ , , z, u ) | w εj,ℓ ( z ′ , , z, u ) dudz ′ = Z C ν Z R | K m j,ℓ (∆ (1) / |T | ,i T ) ( z ′ , u, z, | w εj,ℓ ( z ′ , u, z, dudz ′ ≤ C min { , j + ℓ } M . Putting the above estimate into (8.31), we see that Z IV z,k (cid:12)(cid:12)(cid:12)Z A g ′ K m j,ℓ (∆ (1) / |T | ,i T ) ( g − g ′ ) η ℓ ( i T )( a R )( g ) dg (cid:12)(cid:12)(cid:12) dg ′ ≤ C ( j + ℓ ) ν h k r ∗ (2 ( j + ℓ ) / k r ∗ ) − ν (1+ ε ) min { , j + ℓ } M Z R Z | z |≤ (cid:18)Z R ℓ (1 + 2 ℓ | u ′ − u − S ( z ′ , z ) | ) ε du ′ (cid:19) | η ℓ ( i T )( a R )( g ) | du dz ≤ C ( j + ℓ ) ν h k r ∗ (2 ( j + ℓ ) / k r ∗ ) − ν (1+ ε ) min { , j + ℓ } M k η ℓ ( i T )( a R ) k L . (8.32)86utting the above estimate (8.32) into (8.30), we deduce thatIV ≤ C X k ≥ X j,ℓ ( j + ℓ ) ν h k r ∗ (2 ( j + ℓ ) / k r ∗ ) − ν (1+ ε ) min { , j + ℓ } M k η ℓ ( i T )( a R ) k L / | k r ∗ | (2 ν +1) / ≤ C (cid:18) r ∗ (cid:19) ε | R | / k a R k L . (8.33)Combining estimates (8.23), (8.26), (8.28), (8.29), (8.33), the proof of the third part is complete andso is the proof of Proposition 1.6. Acknowledgements:
The authors acknowledge support from the Australian Research Council, research grants DP170103025and DP170101060. We also thank Alessio Martini, Jill Pipher and Fulvio Ricci for helpful commentson this work.
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E-mail : [email protected] E-mail : [email protected] Department of Mathematics and Statistics, Macquarie University, NSW, 2109, Australia
E-mail : [email protected] School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia
E-mail : [email protected]@unsw.edu.au