A sharp variant of the Marcinkiewicz theorem with multipliers in Sobolev spaces of Lorentz type
aa r X i v : . [ m a t h . C A ] A ug A SHARP VARIANT OF THE MARCINKIEWICZ THEOREMWITH MULTIPLIERS IN SOBOLEV SPACES OF LORENTZ TYPE
LOUKAS GRAFAKOS, MIECZYS LAW MASTY LO, AND LENKA SLAV´IKOV ´A
Abstract.
Given a bounded measurable function σ on R n , we let T σ be the oper-ator obtained by multiplication on the Fourier transform by σ . Let 0 < s ≤ s ≤· · · ≤ s n < ψ be a Schwartz function on the real line whose Fourier trans-form b ψ is supported in [ − , − / ∪ [1 / ,
2] and which satisfies P j ∈ Z b ψ (cid:0) − j ξ (cid:1) = 1for all ξ = 0. In this work we sharpen the known forms of the Marcinkiewiczmultiplier theorem by finding an almost optimal function space with the propertythat, if the function( ξ , . . . , ξ n ) n Y i =1 ( I − ∂ i ) si h n Y i =1 b ψ ( ξ i ) σ (2 j ξ , . . . , j n ξ n ) i belongs to it uniformly in j , . . . , j n ∈ Z , then T σ is bounded on L p ( R n ) when | p − | < s and 1 < p < ∞ . In the case where s i = s i +1 for all i , it was proved in[13] that the Lorentz space L s , ( R n ) is the function space sought. In this workwe address the significantly more difficult general case when for certain indices i we might have s i = s i +1 . We obtain a version of the Marcinkiewicz multipliertheorem in which the space L s , is replaced by an appropriate Lorentz spaceassociated with a certain concave function related to the number of terms among s , . . . , s n that equal s . Our result is optimal up to an arbitrarily small power ofthe logarithm in the defining concave function of the Lorentz space. Introduction
Let C ∞ ( R n ) be the space of smooth functions with compact support on R n .Given any function σ in L ∞ ( R n ), we consider the multiplier operator T σ defined forall f ∈ C ∞ ( R n ) by T σ f ( x ) = Z R n b f ( ξ ) σ ( ξ ) e πix · ξ dξ, x ∈ R n . As usual, here and in the sequel, b f denotes the Fourier transform of f given by b f ( ξ ) = Z R n f ( x ) e − πix · ξ dx, ξ ∈ R n . The theory of multipliers is vast and extensive but basic material about them canbe found in [18], [11] and [22].A classical problem in harmonic analysis is to find good sufficient conditions onfunctions σ guaranteeing that T σ extends to a bounded operator on L p ( R n ) for some1 < p < ∞ . If this is the case, then σ is called an L p Fourier multiplier. This problem
Mathematics Subject Classification.
Primary 42B15. Secondary 42B25.
Key words and phrases.
Multiplier theorems, Sobolev spaces, Lorentz spaces.The first author acknowledges the support of the Simons Foundation. The second author wassupported by the National Science Centre, Poland, Grant no. 2019/33/B/ST1/00165. The thirdauthor was supported by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)under Germany’s Excellence Strategy - GZ 2047/1, Projekt-ID 390685813. has a long history going back to Bernstein, Hardy, Weyl, Marcinkiewicz, Mikhlinand was studied in the sixties by several mathematicians including Calder´on [4],Hirschman [17], H¨ormander [18], de Leeuw [23], Carleson and Sj¨olin [8].The significance of the multiplier problem lies in the fact that many classical L p boundedness problems in analysis can be described in terms of Fourier multipliers.Several conditions on σ are known to imply boundedness for T σ on L p ( R n ). We arenot going into a complete historical overview of multiplier theory, but we focus onversions of the Marcinkiewicz multiplier theorem. We start with the classical resultof Marcinkiewicz [25], first proved in the context of two-dimensional Fourier series,which basically says (in n dimensions) that if for all α j ∈ { , } (1.1) (cid:12)(cid:12)(cid:12) ∂ α ξ · · · ∂ α n ξ n m ( ξ ) (cid:12)(cid:12)(cid:12) ≤ C α | ξ | − α · · · | ξ n | − α n , ξ j = 0 when α j = 1 , then T m is bounded on L p ( R n ) for all p ∈ (1 , ∞ ).In order to fine-tune this theorem we discuss a version of it where the derivatives α j could be fractional. To describe this we introduce a Schwartz function ψ on R whose Fourier transform is supported in [ − , − / ∪ [1 / ,
2] and which satisfies P j ∈ Z b ψ (2 − j ξ ) = 1 for all ξ = 0. We then define a function Ψ on R n such that(1.2) b Ψ = b ψ ⊗ · · · ⊗ b ψ | {z } n times . Here, ( g ⊗ · · · ⊗ g n )( x , . . . , x n ) := g ( x ) · · · g n ( x n ) , ( x , . . . , x n ) ∈ R n , stands for the tensor product of functions g j : R → C , 1 ≤ j ≤ n . We use thefollowing notation for the differential operatorΓ( s , . . . , s n ) := ( I − ∂ ) s / · · · ( I − ∂ n ) s n / , where ∂ j denotes differentiation in the j th variable. We also introduce the multi-dilation operator D j ,...,j n g ( ξ , . . . , ξ n ) := g (2 j ξ , . . . , j n ξ n ) , ( ξ , . . . , ξ n ) ∈ R n , where g is a function on R n and j , . . . , j n ∈ Z .When 0 < /r < s ≤ · · · ≤ s n <
1, it was shown in [15] that if(1.3) sup j ,...,j n ∈ Z (cid:13)(cid:13)(cid:13) Γ( s , . . . , s n ) (cid:2) b Ψ D j ,...,j n σ (cid:3)(cid:13)(cid:13)(cid:13) L r ( R n ) < ∞ , then T σ maps L p ( R n ) to L p ( R n ) when | p − | < s . Earlier versions of this resultwere provided by Carbery [5], who considered the case in which the multiplier liesin a product-type L -based Sobolev space, and Carbery and Seeger [6, Remarkafter Prop. 6.1], who considered the case s = · · · = s n > | p − | = r . Thepositive direction of Carbery and Seeger’s result in the range | p − | < r alsoappeared in [7, Condition (1.4)]; note that in these cases the range is expressedin terms of the integrability of the multiplier and not in terms of its smoothness.Alternative improvements and variants of the Marcinkiewcz multiplier theorem werealso proved by Coifman, Rubio de Francia and Semmes [9] and Tao and Wright [30].An extension of the Marcinkiewicz multiplier theorem to general Banach spaces wasobtained by Hyt¨onen [19].A weakening of the condition in (1.3) was provided in [13], where the L r spacewas replaced by the locally larger Lorentz space L /s , ( R n ). But this was achieved ARCINKIEWICZ MULTIPLIER THEOREM 3 under the additional hypothesis that s < s < · · · < s n ; the case n = 2 was firstproved in [12]. In this paper we deal with the more complicated case when a streakof s j ’s could be identical. In this case, the Lorentz-space estimate from [13] fails(see Example 4.4 below for the proof of this assertion). Nevertheless, we show thata limiting version of the Marcinkiewicz multiplier theorem can still be obtained. Weachieve this goal by enlarging the original Lorentz space L /s , ( R n ) by inserting inthe defining function a certain power of the logarithm. The class of function spacesthat is suitable for solving this problem is described below. Given a concave function ϕ : R + → R + that is positive on (0 , ∞ ) and satisfies ϕ (0) = 0, we define the Lorentzspace Λ ϕ on R + to be the space of all measurable functions f on R + for which k f k Λ ϕ := Z ∞ f ∗ ( t ) dϕ ( t ) < ∞ , where f ∗ denotes the non-increasing rearrangement of f . If 1 ≤ p < ∞ and ϕ ( t ) = t /p for all t ≥
0, then we recover the classical Lorentz space L p, .For s ∈ (0 ,
1) and β ∈ R , we consider a concave function φ s,β such that(1.4) φ s,β ( t ) ≈ t s log β (cid:0) e + t (cid:1) , t > . The main result of this paper is the following theorem.
Theorem 1.1.
Let < p < ∞ and Ψ be as in (1.2) . Let < s ≤ s ≤ · · · ≤ s n < and assume that there are exactly d numbers among s , . . . , s n that equal s . Inaddition, assume that s > | /p − / | . If a function σ ∈ L ∞ ( R n ) satisfies (1.5) K := sup j ,...,j n ∈ Z (cid:13)(cid:13)(cid:13) Γ ( s , . . . , s n ) (cid:2) b Ψ D j ,...,j n σ (cid:3)(cid:13)(cid:13)(cid:13) Λ φs ,d ( R n ) < ∞ , then there is constant C = C ( s , . . . , s n , p, n, d, ψ ) such that, for every f ∈ C ∞ ( R n ) ,we have (1.6) k T σ f k L p ( R n ) ≤ CK k f k L p ( R n ) . Thus, T σ admits a bounded extension from L p ( R n ) to L p ( R n ) with the same bound. Naturally, the theorem remains invariant under any permutation of the variables.It was only stated in the case where the index s j corresponds to variable ξ j forsimplicity. We also point out that the power d of the logarithm in condition (1.5)can be slightly lowered if we allow it to depend on s ; on this improvement seeRemark 6.7. The results contained in Theorem 1.1 and Remark 6.7 inspire thefollowing speculation related to the optimal power of the logarithm. Conjecture 1.2.
Let p , Ψ , s , . . . , s n be as in Theorem 1.1. If a function σ in L ∞ ( R n ) satisfies (1.7) K ′ := sup j ,...,j n ∈ Z (cid:13)(cid:13)(cid:13) Γ ( s , . . . , s n ) (cid:2) b Ψ D j ,...,j n σ (cid:3)(cid:13)(cid:13)(cid:13) Λ φs , (1 − s d ( R n ) < ∞ , then there is constant C = C ( s , . . . , s n , p, n, d, ψ ) such that, for every f ∈ C ∞ ( R n ) ,we have k T σ f k L p ( R n ) ≤ CK ′ k f k L p ( R n ) . We recall that k T σ k L p ( R n ) → L p ( R n ) & k σ k L ∞ ( R n ) , and the appearance of the spaceΛ φ s , (1 − s d ( R n ) in Conjecture 1.2 is motivated by the fact that among all rearrangement-invariant spaces E ( R n ) satisfying the Sobolev-type embedding k σ k L ∞ ( R n ) . k Γ( s , . . . , s n ) σ k E ( R n ) , LOUKAS GRAFAKOS, MIECZYS LAW MASTY LO, AND LENKA SLAV´IKOV ´A Λ φ s , (1 − s d ( R n ) is locally the largest one, see Proposition 4.2 below. Our emphasison the local behavior of the function spaces involved when investigating optimalityquestions is then justified by the local nature of the condition (1.5). We point outthat while the validity of Conjecture 1.2 remains an open problem, we are able toshow that condition (1.7) is sufficient for the L p boundedness of the operator T σ when the power (1 − s ) d of the logarithm is replaced by (1 − s ) d + ε for some ε >
0, assuming that s ≤ /
2; this is the content of Remark 6.7 below.Throughout the paper we use standard notation. Given two nonnegative functions f and g defined on the same set A , we write f . g , if there is a constant c > f ( x ) ≤ cg ( x ) for all x ∈ A , while f ≈ g means that both f . g and g . f hold. If X and Y are Banach spaces, then X ֒ → Y means that X ⊂ Y and theinclusion map is continuous. If X and Y are Banach spaces, then we write X = Y if X ֒ → Y and Y ֒ → X . The measure space of all Lebesgue’s measurable subsetsof R n equipped with Lebesgue measure λ n is denoted by ( R n , λ n ). For simplicityof notation, λ denotes the Lebesgue measure restricted to Lebesgue’s measurablesubset of R + := [0 , ∞ ). We use C to describe an inessential constant that may varyfrom occurrence to occurrence.2. Background material
Let (Ω , µ ) := (Ω , Σ , µ ) be a σ -finite measure space and let L ( µ ) denote the spaceof all (equivalence classes) of scalar valued (real or complex) Σ-measurable functionson (Ω , µ ) (on Ω for short) that are finite µ -a.e. A Banach space X ⊂ L ( µ ) is saidto be a Banach function space over Ω if for all f, g ∈ L ( µ ) with | g | ≤ | f | µ -a.e. and f ∈ X , one has g ∈ X and k g k X ≤ k f k X . The K¨othe dual space X ′ of a Banachfunction space X on Ω is a Banach function space of those f ∈ L ( µ ) for which k f k X ′ := sup (cid:8) R Ω | f g | dµ : k g k X ≤ (cid:9) is finite.Given f ∈ L ( µ ), its distribution function is defined by µ f ( τ ) = µ ( { x ∈ Ω : | f ( x ) | > τ } ), τ >
0, and its nonincreasing rearrangement by f ∗ ( t ) = inf { τ ≥ µ f ( τ ) ≤ t } , t ≥
0. A Banach function space E is called a rearrangement-invariant (r.i.) space if k f k E = k g k E whenever µ f = µ g and f ∈ E .Let E be an r.i. space on R + and let (Ω , µ ) be a measure space. Then we definethe r.i. space E (Ω) on Ω to be the space of all f ∈ L ( µ ) such that f ∗ ∈ E with k f k E (Ω) = k f ∗ k E . Many properties of r.i. spaces can be expressed in terms ofconditions on their Boyd indices. Recall that for any r.i. space E on R + , we definethe dilation operators σ s for 0 < s < ∞ by σ s f ( t ) = f ( t/s ) , f ∈ E, t ≥ . Since s
7→ k σ s k E = sup k f k E ≤ k σ s f k E is a finite submultiplicative function on (0 , ∞ ),the Boyd indices given by α E := lim s → log k σ s k E log s , β E := lim s →∞ log k σ s k E log s are well defined and satisfy 0 ≤ α E ≤ β E ≤ P be the set of functions ϕ : R + → R + that are concave,positive on (0 , ∞ ) and ϕ (0) = 0.Given ϕ ∈ P , the Lorentz space Λ ϕ on R + consists of all f ∈ L ( λ ) such that k f k Λ ϕ := Z ∞ f ∗ ( t ) dϕ ( t ) = ϕ (0+) f ∗ (0+) + Z ∞ f ∗ ( t ) ϕ ′ ( t ) dt , ARCINKIEWICZ MULTIPLIER THEOREM 5 where ϕ ′ is the derivative of ϕ , which exists except at a countable set. We note thatthe functional k · k Λ ϕ induced by an increasing function ϕ : R + → R + is a norm ifand only if ϕ is concave and ϕ (0) = 0 [24]. We note that Λ ϕ is a separable space ifand only if ϕ (0+) = 0 and ϕ (+ ∞ ) := lim t →∞ ϕ ( t ) = ∞ .Let Q be the set of all quasi-concave functions ϕ : R + → R + , that is, of all positivefunctions ϕ on (0 , ∞ ) such that ϕ ( s ) ≤ max { , s/t } ϕ ( t ) for all s, t >
0. Note that,for any ϕ ∈ Q , the function ϕ ∗ given by ϕ ∗ ( t ) := t/ϕ ( t ) for all t > concave majorant defined by e ϕ ( t ) = inf s> (cid:16) ts (cid:17) ϕ ( s ) , which satisfies ϕ ( t ) ≤ e ϕ ( t ) ≤ ϕ ( t ) for all t > ϕ ∈ Q , the Marcinkiewicz space M ϕ on R + is the r.i. space of all f ∈ L ( λ ) equipped with the norm k f k M ϕ := sup t> ϕ ( t ) f ∗∗ ( t ) , where f ∗∗ ( t ) := t R t f ∗ ( s ) ds for all t > ϕ ( R n ) and the Marcinkiewicz space M ϕ ( R n )over the measure space ( R n , λ n ). We will use the K¨othe duality between Lorentzand Marcinkiewicz spaces, which states that for any ϕ ∈ P with ϕ (0+) = 0, wehave Λ ϕ ( R n ) ′ = M ϕ ∗ ( R n )with equality of norms. As a consequence, we have the following variant of H¨older’sinequality (see, e.g., [21, Theorem 5.2] or [2, Chapter 1, Theorem 2.4]):(2.1) Z R n | f g | dλ n ≤ k f k Λ ϕ ( R n ) k g k M ϕ ∗ ( R n ) . In what follows, for simplicity of notation, we often write Λ ϕ and M ϕ for shortinstead of Λ ϕ ( R n ) and M ϕ ( R n ).We will also consider a class B of all measurable functions ψ : R + → R + such thatthe function m ψ is finite and measurable, where m ψ ( t ) := sup s> ψ ( st ) ψ ( s ) , t > . The lower and the upper index of a function ψ ∈ B are defined by γ ψ = lim t → log m ψ ( t )log t , δ ψ = lim t →∞ log m ψ ( t )log t . We have −∞ < γ ψ ≤ δ ψ < ∞ (see [21, Section 2, p. 53]). Note that ϕ, ψ ∈ B with ϕ ≈ ψ implies γ ϕ = γ ψ and δ ϕ = δ ψ .In the sequel we will use the following properties without any references:(i) Every function ψ ∈ B with 0 < γ ψ ≤ δ ψ < concave majorant (see [21, Corollary 2, p. 55]).(ii) If ϕ ∈ P with γ ϕ >
0, then it follows from [21, Lemma 2.1.4] that(2.2) ϕ ( t ) ≈ Z t ϕ ( s ) s ds, t > . LOUKAS GRAFAKOS, MIECZYS LAW MASTY LO, AND LENKA SLAV´IKOV ´A
In particular this implies that, k f k Λ ϕ ≈ Z ∞ f ∗ ( t ) ϕ ( t ) t dt, f ∈ Λ ϕ ( R n ) , up to multiplicative constants depending only on ϕ .(iii) If ϕ ∈ Q with δ ϕ <
1, then γ ϕ ∗ = 1 − δ ϕ > k f k M ϕ ≈ sup t> ϕ ( t ) f ∗ ( t ) , f ∈ M ϕ ( R n ) . (iv) If ϕ ∈ P with 0 < γ ϕ ≤ δ ϕ <
1, then the Lorentz space Λ ϕ ( R n ) is separable (by ϕ (0+) = 0 and ϕ (+ ∞ ) = ∞ ). In particular, it follows that the space C ∞ ( R n ) isdense in Λ ϕ ( R n ).Throughout the paper we consider two families of special concave functions asso-ciated with indices s ∈ (0 ,
1) and β ∈ R . The function φ s,β was defined in (1.4). Inaddition, we let ω s,β be a concave function satisfying ω s,β ( t ) ≈ t s log β ( e + t ) , t > . Basic properties of the functions φ s,β and ω s,β are summarized in the followingproposition. Proposition 2.1.
Given s > , β ∈ R , consider the functions φ and ω defined by φ ( t ) := t s log β (cid:0) e + t (cid:1) and ω ( t ) := t s log β ( e + t ) for all t > . Then φ, ω ∈ B with γ φ = δ φ = s and γ ω = δ ω = s . If, in addition, s ∈ (0 , , then φ and ω are equivalentto their concave majorant denoted by φ s,β and ω s,β , respectively.Proof. We first focus on the case when β ≥
0. We observe that, for all a, b ≥ e + ab ) < log[( e + a )( e + b )] = log( e + a ) + log( e + b ) ≤ e + a )] [log( e + b )] . This shows that, for C = 1 / log β ( e + 1), we have Cφ ( t ) ≤ m φ ( t ) = sup r> φ ( rt ) φ ( r ) ≤ β φ ( t ) , t > , and so m φ ≈ φ . Since φ is continuous, it follows that φ ∈ B and we have γ φ = lim t → log m φ ( t )log t = lim t → log φ ( t )log t = s + β lim t → log(log( e + t − ))log t = s and δ φ = s + β lim t →∞ log(log( e + t − ))log t = s . Similarly, we deduce that ω ∈ B with m ω ≈ ω and γ ω = δ ω = s .Notice that m φ β ( t ) = m ω − β ( t ) and m ω β ( t ) = m φ − β ( t ) for t > β ∈ R , wherewe set φ β := φ and ω β := ω . These equalities yield the conclusion for β < − β >
0. If s ∈ (0 , (cid:3) We will need the following lemma. For completeness we include a proof.
ARCINKIEWICZ MULTIPLIER THEOREM 7
Lemma 2.2.
Let ψ ∈ B with δ ψ < p for some < p ≤ . Then there existsa constant C = C ( ψ, p ) > such that, for any h ∈ L ( λ n ) , we have Z ∞ h ∗∗ ( t ) p ψ ( t ) t dt ≤ C Z ∞ h ∗ ( t ) p ψ ( t ) t dt . Proof.
Observe that for any t >
0, we have an obvious estimate h ∗∗ ( t ) = Z h ∗ ( ts ) ds = ∞ X k =1 Z − k +1 − k h ∗ ( ts ) ds ≤ ∞ X k =1 − k h ∗ ( t/ k ) . Combining with subadditivity of the function t t p , defined on R + , yields Z ∞ h ∗∗ ( t ) p ψ ( t ) t dt ≤ ∞ X k =1 − kp Z ∞ h ∗ ( t/ k ) p ψ ( t ) t dt = ∞ X k =1 − kp Z ∞ h ∗ ( t ) p ψ (2 k t ) t dt ≤ C ( p, ψ ) Z ∞ h ∗ ( t ) p ψ ( t ) t dt , where C ( p, ψ ) := P ∞ k =1 − kp m ψ (2 k ).We complete the proof by showing that C ( p, ψ ) < ∞ . To see this observe that by δ ψ < p , we can find ε > α := p − δ ψ − ε >
0. It follows from the definitionof δ ψ that there is an integer k = k ( ε ) > m ψ (2 k ) ≤ k ( δ ψ + ε ) for each k ≥ k and hence ∞ X k = k − kp m ψ (2 k ) ≤ ∞ X k = k − kα < ∞ . This concludes the proof. (cid:3)
We now prove the following result on boundedness of the Fourier transform be-tween corresponding Lorentz spaces on R n . Before doing so, we point out thatvarious variants of the Hausdorff-Young inequality in the setting of Lorentz spacesare available in the literature (see, e.g., [1], [27], [28]) but the result below appearsto be new. Lemma 2.3.
Let ϕ ∈ P such that / < γ ϕ ≤ δ ϕ < . Then there exists a constant C > such that, for any f ∈ Λ ϕ , we have k b f k Λ ψ ≤ C k f k Λ ϕ , where ψ ( t ) := tϕ (1 /t ) for all t > .Proof. Since ϕ is concave, it is easy to check that ψ is a concave function on (0 , ∞ ).Clearly, m ψ ( t ) = t m ϕ (1 /t ) for all t > γ ψ = 1 − δ ϕ and δ ψ = 1 − γ ϕ . Hence it follows by assumption on indices of ϕ that 0 < γ ψ ≤ δ ψ < / D > f ∈ L ( R n ) + L ( R n ), we have Z t ( b f ) ∗ ( s ) ds ≤ D Z t (cid:18) Z s f ∗ ( τ ) dτ (cid:19) ds . LOUKAS GRAFAKOS, MIECZYS LAW MASTY LO, AND LENKA SLAV´IKOV ´A
Combining this estimate with( b f ) ∗ ( t ) ≤ t / (cid:18) Z t (( b f ) ∗ ( s )) ds (cid:19) , t > , we obtain that for any simple function f ∈ Λ ϕ , k b f k Λ ψ ≤ D Z ∞ (cid:18) t Z t (cid:18) Z s f ∗ ( τ ) dτ (cid:19) ds (cid:19) ψ ( t ) t dt = Z ∞ g ∗∗ ( t ) ψ ( t ) t dt , where g is given by g ( t ) := (cid:18) Z t f ∗ ( τ ) dτ (cid:19) , t > . Clearly, g is non-negative and nonincreasing and so it follows from Lemma 2.2 (by δ ψ < /
2) that there exists
C > Z ∞ g ∗∗ ( t ) ψ ( t ) t dt ≤ C Z ∞ g ( t ) ψ ( t ) t dt . In consequence, we obtain k b f k Λ ψ . Z ∞ (cid:18) Z t f ∗ ( s ) ds (cid:19) ψ ( t ) t dt = Z ∞ f ∗∗ ( t − ) ψ ( t ) t dt = Z ∞ f ∗∗ ( t ) ψ ( t − ) dt = Z ∞ f ∗∗ ( t ) ϕ ( t ) t dt . Z ∞ f ∗ ( t ) ϕ ( t ) t dt , where the last estimate follows by Lemma 2.2 with p = 1 (by δ ϕ < γ ϕ > k f k Λ ϕ ≈ Z ∞ f ∗ ( t ) ϕ ( t ) t dt . Thus the required estimate follows by density of simple functions in the Lorentzspace Λ ϕ . (cid:3) Preliminary results
In this section we prove various auxiliary results that will be crucial in the proofof Theorem 1.1. We start with an estimate for an integral.
Lemma 3.1. If k ≥ , α , . . . , α k > , r , . . . , r k > are such that ( α − /r ≤ ( α − /r ≤ · · · ≤ ( α k − /r k and a > , then (3.1) Z · · · Z u ,...,u k ≥ u r ··· u rkk >a u − α · · · u − α k k du · · · du k ≈ a − α r [log( e + a )] d ′ , up to multiplicative constants independent of a . Here, d ′ is the number of elementsin { ( α − /r , ( α − /r , . . . , ( α k − /r k } that are equal to ( α − /r . ARCINKIEWICZ MULTIPLIER THEOREM 9
Proof.
To prove (3.1) we proceed by induction. First we verify the case k = 2. In thiscase the u integral is over the region u ≥ max (cid:8) , ( au − r ) /r (cid:9) and so evaluatingthe u integral gives(3.2) Z Z u ,u ≥ u r u r >a u − α u − α du du = C Z ∞ u =1 u − α max (cid:8) , ( au − r ) /r (cid:9) − α du . If the maximum equals 1, then the integral is over the region a /r ≤ u < ∞ andthe u integration produces Ca /r (1 − α ) ≤ Ca /r (1 − α ) . Thus, the correspondingpart of (3.2) is bounded from above by the right-hand side of (3.1), and if a ∈ (1 , au − r ) /r , then the integral is over the region 1 ≤ u ≤ a /r and so the corresponding part of (3.2) becomes Z a r u =1 u − α ( au − r ) (1 − α ) /r du = a − α r Z a r u =1 u − α − r /r (1 − α )2 du . Now if ( α − /r > ( α − /r then this is bounded from above by the right-handside of (3.1) with d ′ = 0 and if ( α − /r = ( α − /r then the same estimateholds with d ′ = 1. In addition, one has the corresponding lower bound if a > k = 2.Assume by induction that (3.1) holds for an integer k − k ). Then Z · · · Z u ,...,u k ≥ u r ··· u rkk >a u − α · · · u − α k k du · · · du k = Z ∞ u =1 · · · Z ∞ u k =1 R ∞ u = L u − α du u α k k · · · u α du k · · · du , where L = max { , ( au − r · · · u − r k k ) /r } . As α >
1, the u integral is convergent andthe preceding expression equals(3.3) c Z ∞ u =1 · · · Z ∞ u k =1 max n , ( au − r · · · u − r k k ) r o − α du k u α k k · · · du u α . The part of the integral in (3.3) over the set where the maximum equals 1 is(3.4) c Z · · · Z u ,...,u k ≥ u r ··· u rkk >a u − α k k · · · u − α du k · · · du ≈ a − α r log d ′′ ( e + a ) , where the equivalence holds by the induction hypothesis and d ′′ is the number ofelements in { ( α − /r , . . . , ( α k − /r k } that are equal to ( α − /r . Note thatif ( α − /r < ( α − /r , then the expression on the right in (3.4) is boundedfrom above by(3.5) Ca − α r log d ′ ( e + a ) . Now if ( α − /r = ( α − /r , then we have d ′ = d ′′ +1 and then the expression onthe right in (3.4) is also bounded by (3.5). In addition, we also have the correspond-ing lower bound in both cases within the range a ∈ (1 , au − r · · · u − r k k ) r . It can be expressed as(3.6) ca − α r Z · · · Z u ,...,u k ≥ u r ··· u rkk ≤ a u rkr ( α − − α k k · · · u r r ( α − − α du k · · · du . First, we observe that we have the following upper bound for (3.6): ca − α r Z a r · · · Z a rk u rkr ( α − − α k k · · · u r r ( α − − α du k · · · du ≤ Ca − α r log d ′ ( e + a ) , where the logarithm appears exactly when r k r = α k − α − ( d ′ times) and the remainingintegrals produce a constant. Conversely, if a > ca − α r Z a k − r · · · Z a k − rk u rkr ( α − − α k k · · · u r r ( α − − α du k · · · du ≈ Ca − α r log d ′ ( e + a ) . The claim follows. (cid:3)
We denote by M the strong maximal operator defined at point as the supremumof the averages of a given function over all rectangles with sides parallel to the axesthat contain the point. Then we define M L q ( g )( x , . . . , x n ) = M ( | g | q )( x , . . . , x n ) q , a version of the strong maximal function with respect to an exponent q ∈ (1 , ∞ ). Lemma 3.2.
Let < /q < s ≤ s ≤ · · · ≤ s n < . Suppose that exactly d of thenumbers s , . . . , s n are equal to s , where ≤ d ≤ n − . Then for g in L loc ( R n ) with M L q ( g )(0) = 1 and a > we have (3.7) (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) y ∈ R n \ [ − , n : | g ( y ) | Q ni =1 (1 + | y i | ) s i > a (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Ca − s log d (cid:16) e + 1 a (cid:17) . Proof.
For j , . . . , j n nonnegative integers define R j ,...,j n = ( ( y , . . . , y n ) ∈ R n : ( j i < | y i | ≤ j i +1 if j i ≥ | y i | ≤ j i = 0 , ≤ i ≤ n ) and notice that the family of rectangles R j ,...,j n is a tiling of R n when j , . . . , j n runover all nonnegative integers.In the sequel we denote by y the vector ( y , . . . , y n ). For a > j , . . . , j n nonnegative integers, we have |{ y ∈ R j ,...,j n : | g ( y ) | > a }| ≤ a q Z R j ,...,jn | g ( y ) | q dy ≤ a − q j + ··· + j n +2 n since we are assuming that M L q ( g )(0) = 1. Thus, in view of the trivial estimate | R j ,...,j n | ≤ j + ··· + j n +2 n , we obtain(3.8) |{ y ∈ R j ,...,j n : | g ( y ) | > a }| ≤ n j + ··· + j n min (cid:8) , a − q (cid:9) . ARCINKIEWICZ MULTIPLIER THEOREM 11
It follows from (3.8) that, for all j , . . . , j n ≥
0, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) y ∈ R j ,...,j n : | g ( y ) | (1 + | y | ) s · · · (1 + | y n | ) s n > a (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) ≤ n j + ··· + j n min (cid:8) , ( a j s + ··· j n s n ) − q (cid:1) . (3.9)We let g = gχ R n \ R ,..., . Using (3.9), we get that (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) y ∈ R n : | g ( y ) | (1 + | y | ) s · · · (1 + | y n | ) s n > a (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ∞ X j ,...,j n =0 j + ··· + j n > (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) y ∈ R j ,...,j n : | g ( y ) | (1 + | y | ) s · · · (1 + | y n | ) s n > a (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ∞ X j ,...,j n =0 j + ··· + j n > j + ··· + j n +2 n min (cid:8) , a − q (2 j s + ··· + j n s n ) − q (cid:9) ≤ n + s q + ··· + s n q Z · · · Z [0 , ∞ ) n \ [0 , n min (cid:8) , a − q n Y ρ =1 max { , t s ρ ρ } − q (cid:9) dt · · · dt n =: 2 n + s q + ··· + s n q I ( a, n ) , (3.10)where the last inequality follows from the monotonicity of the integrand. Let S bethe set of all ( t , . . . t n ) ∈ [0 , ∞ ) n \ [0 , n such that(3.11) a max { , t s } · · · max { , t s n n } ≤ . If S is nonempty, then we must have a ≤
1. Let us fix a two-set partition I = { i , . . . , i m } and J = { j , . . . , j k } of { , , . . . , n } . We split S as a union of sets S I,J (ranging over all such pairs of partitions) for which(3.12) ( t , . . . , t n ) ∈ S I,J ⇐⇒ t i ≤ i ∈ I and t j > j ∈ J .Then the n -dimensional measure | S I,J | of S I,J is at most the k -th dimensional mea-sure of S kI,J = n ( t j , . . . , t j k ) : t s j j · · · t s jk j k ≤ a o ∩ [1 , ∞ ) k , as the vector of the remaining m coordinates is contained in the cube [0 , m whichhas m -th dimensional measure equal to 1. Let us assume, without loss of generality,that s j ≤ s j ≤ · · · ≤ s j k (i.e., j < j < · · · < j k ).We make the following observation: if ( t j , . . . , t j k ) ∈ S kI,J , then1 ≤ t j i ≤ a − s , ≤ i ≤ k . Indeed, as all t j i ≥
1, we have 1 ≤ t s ji j i ≤ t s j j · · · t s jk j k ≤ a − , which implies that1 ≤ t j i ≤ a − /s ji ≤ a − /s . Thus we conclude that | S kI,J | ≤ Z a − s t j =1 · · · Z a − s t jk =1 (cid:12)(cid:12)(cid:12)n t j : 1 ≤ t j ≤ a − sj t − sj sj j · · · t − sjksj j k o(cid:12)(cid:12)(cid:12) dt j k · · · dt j ≤ a − sj χ a ≤ Z a − s t j =1 · · · Z a − s t jk =1 t − sj sj j · · · t − sjksj j k dt j k · · · dt j ≤ Ca − sj χ a ≤ log d ′ (cid:16) a (cid:17) s ≤ C ′ a − s χ a ≤ log d (cid:16) e + 1 a (cid:17) , where d ′ is the number of elements of the set { s j , . . . , s j k } that are equal to s j .The integrals associated with these variables produce a logarithm, while all otherintegrals are convergent on [1 , ∞ ). The last inequality holds independently of therelationship between d and d ′ if s j > s , while if s j = s then it is satisfiedsince d ′ ≤ d . Summing over all partitions ( I, J ) of { , , . . . , n } yields the requiredestimate for I ( a, n ), defined in (3.10), whenever (3.11) holds.Now let S ′ be the set of all ( t , . . . t n ) ∈ [0 , ∞ ) n \ [0 , n such that(3.13) a max { , t s } · · · max { , t s n n } > . Then S ′ is complementary to S in [0 , ∞ ) n \ [0 , n . Writing S ′ as a union of sets S ′ I,J over all partitions (
I, J ) of { , , . . . , n } as in (3.12), matters reduce to estimatingthe integral(3.14) 1 a q Z · · · Z t j ,...,t jk ≥ t sj j ··· t sjkjk > a ( t s j j · · · t s jk j k ) − q dt j · · · dt j k for each subset { j , . . . , j k } of { , . . . , n } . Now if a >
1, the integral in (3.14) is overthe set [1 , ∞ ) k and, as s j q > , . . . , s j k q >
1, the expression in (3.14) is bounded by
C a − q χ a> ≤ C a − s , since q > /s . So we focus attention to the case a ≤ s j ≤ s j ≤ · · · ≤ s j k . To estimate (3.14) weuse Lemma 3.1. Inequality (3.1) in this lemma implies that, if d ′ is the number ofterms in { s j , . . . , s j k } that are equal to s j , then (3.14) is bounded by Ca − sj log d ′ (cid:0) e + a − (cid:1) ≤ Ca − s log d (cid:0) e + a − (cid:1) , where the last inequality is due to the fact that either s j = s and d ′ ≤ d , or s j < s (as a < I, J ) of { , , . . . , n } yields the required estimatefor I ( a, n ), defined in (3.10), whenever (3.13) holds. This completes the proof of(3.7). (cid:3) Corollary 3.3.
Let < /q < s ≤ s ≤ · · · ≤ s n < with exactly d numbersamong s , . . . , s n being equal to s . For g in L loc ( R n ) with M L q ( g )(0) = 1 we have (3.15) (cid:18) | g ( y , . . . , y n ) | χ R n \ [ − , n Q ni =1 (1 + | y i | ) s i (cid:19) ∗ ( t ) ≤ Cω s , − s d ( t ) . Proof.
Note that the inverse function to(0 , ∞ ) ∋ a a − s log d (cid:16) e + 1 a (cid:17) , that appears in Lemma 3.2, is equivalent to t t − s log s d ( e + t ) = (cid:0) ω s , − s d ( t ) (cid:1) − . This proves (3.15). (cid:3)
ARCINKIEWICZ MULTIPLIER THEOREM 13
Proposition 3.4.
Assume that h is supported in the cube [ − , n = Q and that s > /r > /q > . Then (3.16) k h k M ωs , − s d . k h k L r, ∞ . M L q ( h )(0) . Proof.
We first notice that the function h is supported in a set of measure 2 n , andtherefore h ∗ ( t ) = 0 if t > n . Since the function ω s , − s d ( t ) /t is non-increasing, wehave k h k M ωs , − s d = sup t> ω s , − s d ( t ) t Z t h ∗ ( s ) ds = sup t ∈ (0 , n ) ω s , − s d ( t ) t Z t h ∗ ( s ) ds . sup t ∈ (0 , n ) t r t Z t h ∗ ( s ) ds . k h k L r, ∞ . Notice that the first inequality above makes use of the fact that ω s , − s d ( t ) . t /r for t ∈ (0 , n ) as 1 /r < s . This proves the first inequality in (3.16). The second inequal-ity in (3.16) follows from the natural embedding of L q ([ − , n ) in L r, ∞ ([ − , n ), as r < q . (cid:3) Combining the results of Corollary 3.3 and Proposition 3.4 we obtain the following.
Corollary 3.5.
Let < /q < s ≤ s ≤ · · · ≤ s n < with exactly d numbersamong s , . . . , s n being equal to s . For g in L loc ( R n ) with M L q ( g )(0) = 1 we have (3.17) (cid:18) g ( y , . . . , y n ) Q ni =1 (1 + | y i | ) s i (cid:19) ∗ ( t ) ≤ Cω s , − s d ( t ) . Consequently, for any g ∈ L loc ( R n ) and any x = ( x , . . . , x n ) ∈ R n , we have (3.18) (cid:13)(cid:13)(cid:13)(cid:13) g ( x + 2 − j y , . . . , x n + 2 − j n y n ) Q ni =1 (1 + | y i | ) s i (cid:13)(cid:13)(cid:13)(cid:13) M ωs , − s d ( dy ··· dy n ) ≤ C M L q ( g )( x ) , for any j , . . . , j n ∈ Z .Proof. To prove (3.17) we split g = g + g , where g = gχ [ − , n and g = gχ R n \ [ − , n and we apply Corollary 3.3 to g and Proposition 3.4 to g . Now (3.17) applied to g/ M L q ( g )(0) yields (3.18) when x = 0 and j = · · · = j n = 0. The general case of(3.18) can be obtained by a translation and a dilation. (cid:3) A limiting case Sobolev embedding
The following embedding of a Lorentz-Sobolev space into the space of essentiallybounded functions is an important ingredient for the proof of Theorem 1.1.
Proposition 4.1.
Let < s ≤ s ≤ · · · ≤ s n < , where exactly d of the numbers s , . . . , s n are equal to s . Then k f k L ∞ ( R n ) . k Γ( s , . . . , s n ) f k Λ φs , (1 − s d ( R n ) . Proof.
For a given function f , we denote g = Γ( s , . . . , s n ) f . We write f = Γ( − s , . . . , − s n ) g = ( G s ⊗ · · · ⊗ G s n ) ∗ g , where G s is the one-dimensional kernel of ( I − ∂ ) − s/ . We recall the estimates G s ( x ) . | x | s − as x → G s ( x ) . e − c | x | as | x | → ∞ . The H¨older inequality (2.1) yields k f k L ∞ ( R n ) = k ( G s ⊗ · · · ⊗ G s n ) ∗ g k L ∞ ( R n ) ≤ sup (˜ x ,..., ˜ x n ) ∈ R n Z R n G s ( x ) · · · G s n ( x n ) | g (˜ x − x , . . . , ˜ x n − x n ) | dx · · · dx n ≤ k G s ⊗ · · · ⊗ G s n k M φ − s ,d ( s − ( R n ) k g k Λ φs , (1 − s d ( R n ) , as for t > tφ s , (1 − s ) d ( t ) = t − s log d ( s − ( e + t ) = φ − s ,d ( s − ( t ) . It remains to verify that(4.1) G s ⊗ · · · ⊗ G s n ∈ M φ − s ,d ( s − ( R n ) . Given a subset I of { , , . . . , n } , we set J = { , , . . . , n }\ I and write I = ( i , . . . , i k )and J = ( j , . . . , j n − k ) (there is a slight abuse of notation as one of the sets may beempty). Now observe that |{ ( x , . . . , x n ) ∈ R n : G s ( x ) . . . G s n ( x n ) > λ }|≤ X I ⊆{ , ,...,n } |{ ( x i , . . . , x i k ) ∈ ( − , k , ( x j , . . . , x j n − k ) ∈ ( R \ ( − , n − k : | x i | s i − · · · | x i k | s ik − e − c ( | x j | + ··· + | x jn − k | ) > λ }| . X I ⊆{ , ,...,n } |{ ( x i , . . . , x i k ) ∈ (0 , k , ( x j , . . . , x j n − k ) ∈ (1 , ∞ ) n − k : x s i − i · · · x s ik − i k e − c ( x j + ··· + x jn − k ) > λ }| . We denote S I,λ = { ( x i , . . . , x i k ) ∈ (0 , k , ( x j , . . . , x j n − k ) ∈ (1 , ∞ ) n − k : x s i − i · · · x s ik − i k e − c ( x j + ··· + x jn − k ) > λ } . We want to estimate | S I,λ | . To this end, we fix I ⊆ { , , . . . , n } and ( x j , . . . , x j n − k ) ∈ (1 , ∞ ) n − k . Further, for a fixed λ > a = λe c ( x j + ··· + x jn − k ) . If a > |{ ( x i , . . . , x i k ) ∈ (0 , k : x s i − i · · · x s ik − i k > a }| (4.2) = Z · · · Z x i ,...,x ik ∈ (0 , x si − i ··· x sik − ik >a dx i · · · dx i k = Z · · · Z u ,...,u k > u − si ··· u − sikk >a u − . . . u − k du · · · du k . a − − si log d ′ ( e + a ) , where d ′ is the number of elements from the set { s i , . . . , s i k } that are equal to s i . We recall that the last inequality follows from Lemma 3.1. Notice that theestimate (4.2) is true also if a ≤ a − − si log d ′ ( e + a ) . λ − − si log d ′ ( e + λ ) e − c − si ( x j + ··· + x jn − k ) ( x j + · · · + x j n − k ) d ′ ARCINKIEWICZ MULTIPLIER THEOREM 15 . λ − − si log d ′ ( e + λ ) e − c ′ ( x j + ··· + x jn − k ) , where c ′ < c − s i .Thus, if λ > | S I,λ | . Z · · · Z x j ,...,x jn − k > λ − − si log d ′ ( e + λ ) e − c ′ ( x j + ··· + x jn − k ) dx j . . . dx j n − k . λ − − si log d ′ ( e + λ ) . λ − − s log d ( e + λ ) . On the other hand, if λ ≤ |{ ( x i , . . . , x i k ) ∈ (0 , k : x s i − i . . . x s ik − i k e − c ( x j + ··· + x jn − k ) > λ }| (4.3) . min { , λ − − si log d ′ ( e + λ ) e − c ′ ( x j + ··· + x jn − k ) } . If the minimum is equal to 1 then e c ′ ( x j + ··· + x jn − k ) ≤ λ − − si log d ′ ( e + λ ), and so x j + · · · + x j n − k . log( eλ − ). Then the measure of the corresponding part of theset S I,λ is bounded by constant timeslog n − k ( eλ − ) . λ − − s log d ( e + λ ) . Finally, if the minimum in (4.3) is equal to λ − − si log d ′ ( e + λ ) e − c ′ ( x j + ··· + x jn − k ) , then x j + · · · + x j n − k ≥ c ′ log( λ − − si log d ′ ( e + λ ) , and the measure of the corresponding part of the set S I,λ is bounded by constanttimes Z · · · Z x j ,...,x jn − k > x j + ··· + x jn − k ≥ c ′ log( λ − − si log d ′ ( e + λ )) λ − − si log d ′ ( e + λ ) e − c ′ ( x j + ··· + x jn − k ) dx j . . . dx j n − k . Z ∞ c ′ log( λ − − si log d ′ ( e + λ )) λ − − si log d ′ ( e + λ ) e − c ′ r r n − k − dr . λ − − si log d ′ ( e + λ ) Z ∞ c ′ log( λ − − si log d ′ ( e + λ )) e − c ′′ r dr . λ − (1 − c ′′ c ′ ) − si log (1 − c ′′ c ′ ) d ′ ( e + λ ) . λ − − s log d ( e + λ ) . The last inequality holds since c ′′ can be chosen to be any number less than c ′ , andthus 1 − c ′′ /c ′ can be arbitrarily small.Altogether, we proved |{ ( x , . . . , x n ) ∈ R n : G s ( x ) · · · G s n ( x n ) > λ }| . λ − − s log d ( e + λ ) . This yields ( G s ⊗ · · · ⊗ G s n ) ∗ ( t ) . t s − (log( e + 1 /t )) d (1 − s ) , t > , which in turn implies (4.1). (cid:3) Next we show that the previous result is sharp, in the sense that the spaceΛ φ s , (1 − s d is locally the largest rearrangement-invariant space for which Proposi-tion 4.1 holds. Proposition 4.2.
Let < s ≤ s ≤ · · · ≤ s n < , where exactly d of the numbers s , . . . , s n are equal to s . Assume that E is a rearrangement-invariant space suchthat (4.4) k f k L ∞ ( R n ) . k Γ( s , . . . , s n ) f k E ( R n ) . Then E (Ω) ֒ → Λ φ s , (1 − s d (Ω) for all sets Ω ⊆ R n of finite measure.Proof. To prove this claim, we set g = Γ( s , . . . , s n ) f and rewrite inequality (4.4) as(4.5) k ( G s ⊗ · · · ⊗ G s n ) ∗ g k L ∞ ( R n ) . k g k E ( R n ) , where G s is the one-dimensional kernel of ( I − ∂ ) − s/ . For a given (˜ x , . . . , ˜ x n ) ∈ R n ,we have sup k g k E ( R n ) ≤ | ( G s ⊗ · · · ⊗ G s n ) ∗ g (˜ x , . . . , ˜ x n ) | (4.6) = sup k g k E ( R n ) ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z R n G s ( x ) · · · G s n ( x n ) g (˜ x − x , . . . , ˜ x n − x n ) dx . . . dx n (cid:12)(cid:12)(cid:12)(cid:12) = k G s ⊗ · · · ⊗ G s n k E ′ ( R n ) , where E ′ is the K¨othe dual space of E . Thus, (4.5) implies that the function G s ⊗ · · · ⊗ G s n belongs to E ′ ( R n ). We next find a lower bound for the distributionfunction of G s ⊗ · · · ⊗ G s n . Since G s i ( x i ) ≈ x s i − i if 0 < x i < i ∈ { , , . . . , n } ,we obtain for λ > |{ ( x , . . . , x n ) ∈ R n : G s ( x ) · · · G s n ( x n ) > λ }| & |{ ( x , . . . , x n ) ∈ (0 , n : x s − · · · x s n − n > λ }|≈ λ − − s log d ( e + λ ) , where d is the number of elements from the set { s , . . . , s n } that are equal to s .Note that the last equivalence follows by the calculation in (4.2) and by Lemma 3.1.This shows that( G s ⊗ · · · ⊗ G s n ) ∗ ( t ) & t s − log (1 − s ) d (cid:16) e + 1 t (cid:17) , t ∈ (0 , t )for some t >
0. This shows that if g ( t ) := t s − log (1 − s ) d (cid:16) e + 1 t (cid:17) , t > , then the function gχ (0 ,t ) ∈ E ′ := E ′ (0 , ∞ ). To reach the conclusion we observe thatthe embedding E (Ω) ֒ → Λ φ s , (1 − s d (Ω) is, by duality, equivalent to M φ − s , ( s − d (Ω) ֒ → E ′ (Ω). Now, if f is a function satisfying k f k M φ − s , ( s − d (Ω) ≤
1, then f ∗ ( t ) ≤ f ∗∗ ( t ) . g ( t ) , t ∈ (0 , | Ω | ) . Hence k f k E ′ (Ω) = k f ∗ χ (0 , | Ω | ) k E ′ . k gχ (0 , | Ω | ) k E ′ . k gχ (0 ,t ) k E ′ + k χ ( t , | Ω | ) k E ′ . k gχ (0 ,t ) k E ′ ≤ C .
In this chain of inequalities we used the monotonicity and rearrangement-invarianceof the norm in the space E ′ , the fact that the interval ( t , | Ω | ) can be split into a finite ARCINKIEWICZ MULTIPLIER THEOREM 17 number of intervals of length at most t and that the constant function on the inter-val (0 , t ) is bounded from above by a multiple of the function t s − log d (1 − s ) ( e + t ).This completes the proof. (cid:3) Corollary 4.3.
Let < s ≤ s ≤ · · · ≤ s n < , where exactly d of the numbers s , . . . , s n are equal to s . Assume that E is a rearrangement-invariant space suchthat < α E ≤ β E < and (4.7) k T σ f k L p ( R n ) . sup j ,...,j n ∈ Z (cid:13)(cid:13)(cid:13) Γ ( s , . . . , s n ) (cid:2) b Ψ D j ,...,j n σ (cid:3)(cid:13)(cid:13)(cid:13) E ( R n ) k f k L p ( R n ) . Then E (Ω) ֒ → Λ φ s , (1 − s d (Ω) for all sets Ω ⊆ R n of finite Lebesgue measure.Proof. Assume that Φ is a smooth function on R n with compactly supported Fouriertransform and a , . . . , a n are fixed integers. We recall the estimates(4.8) k Γ( s , . . . , s n )[ b Φ F ] k E ( R n ) . k Γ( s , . . . , s n ) F k E ( R n ) and(4.9) k Γ( s , . . . , s n )[ D a ,...,a n F ] k E ( R n ) . k Γ( s , . . . , s n ) F k E ( R n ) for any function F on R n . To verify (4.8) and (4.9) we first observe that they holdin the special case when E = L q , 1 < q < ∞ . Then we choose 1 < q , q < ∞ suchthat 1 /q < α E ≤ β E < /q , and the conclusion follows by interpolating betweenthe L q and L q endpoints via Boyd’s interpolation theorem [3, Theorem 1] (see alsothe beginning of Section 6 for the statement of this theorem).Let us consider testing functions σ of the form(4.10) σ = [( G s ⊗ · · · ⊗ G s n ) ∗ g ] b η, where η is a smooth function on R n satisfying b η = 1 on the cube [7 / , / n and suchthat the support of b η is contained in [3 / , / n . Taking into account the supportproperties of b Ψ we deduce that b Ψ D j ,...,j n σ = 0 unless j i ∈ {− , , } for each i = 1 , , . . . , n . Inequality (4.7) combined with the fact that k T σ k L p ( R n ) → L p ( R n ) & k σ k L ∞ ( R n ) yields(4.11) k σ k L ∞ ( R n ) . sup j ,...,j n ∈{− , , } (cid:13)(cid:13)(cid:13) Γ ( s , . . . , s n ) (cid:2) b Ψ D j ,...,j n σ (cid:3)(cid:13)(cid:13)(cid:13) E ( R n ) . Using (4.8) and (4.9), this implies k σ k L ∞ ( R n ) . k Γ ( s , . . . , s n ) σ k E ( R n ) . An application of (4.10) and (4.8) then gives k [( G s ⊗ · · · ⊗ G s n ) ∗ g ] b η k L ∞ ( R n ) . k g k E ( R n ) . Since b η = 1 on [7 / , / n , the proof of Proposition 4.2 applied with (˜ x , . . . , ˜ x n ) ∈ [7 / , / n yields the conclusion. (cid:3) Example 4.4.
We apply Corollary 4.3 with the Lorentz space E = Λ φ s ,β , where β ∈ R (note that α Λ φs,β = β Λ φs,β = s ). Thus, a necessary condition for inequality(4.12) k T σ f k L p ( R n ) . sup j ,...,j n ∈ Z (cid:13)(cid:13)(cid:13) Γ ( s , . . . , s n ) (cid:2) b Ψ D j ,...,j n σ (cid:3)(cid:13)(cid:13)(cid:13) Λ φs ,β ( R n ) k f k L p ( R n ) to be satisfied is the validity of embedding Λ φ s ,β (Ω) ֒ → Λ φ s , (1 − s d (Ω) for all setsΩ ⊆ R n of finite measure. This is equivalent to the pointwise estimate φ s , (1 − s ) d ( t ) . φ s ,β ( t ) for t near 0 (see, e.g., [26, Theorem 10.3.8]), which in turn yields the explicit necessary condition β ≥ (1 − s ) d . In particular, β = 0 is not allowed unless d = 0,and estimate k T σ f k L p ( R n ) . sup j ,...,j n ∈ Z (cid:13)(cid:13)(cid:13) Γ ( s , . . . , s n ) (cid:2) b Ψ D j ,...,j n σ (cid:3)(cid:13)(cid:13)(cid:13) L s , ( R n ) k f k L p ( R n ) thus fails whenever at least one of the indices s , . . . , s n equals s . On the otherhand, if s ≤ / β > (1 − s ) d , see Remark 6.7 below.5. The core of the proof
In this section we prove Theorem 1.1 in the special case when 1 / < s <
1. Thegeneral case then follows by interpolation; the details can be found in Section 6. Wepoint out that in fact we prove a slightly stronger variant of Theorem 1.1 in thisparticular case; namely, we replace the constant K in (1.5) by the smaller constant e K = sup j ,...,j n ∈ Z (cid:13)(cid:13)(cid:13) Γ ( s , . . . , s n ) (cid:2) b Ψ D j ,...,j n σ (cid:3)(cid:13)(cid:13)(cid:13) Λ φs ,s d ( R n ) . Proof of Theorem . case / < s < . Given a Schwartz function ψ as in thestatement of the theorem, we define a new Schwartz function ψ b ( ψ big) on R asfollows:(5.1) b ψ b ( ξ ) = b ψ ( ξ/
2) + b ψ ( ξ ) + b ψ (2 ξ ) . Then b ψ b is supported in the annulus 1 / < | ξ | < b ψ b = 1 on the support of b ψ .Recalling the definition of Ψ given in (1.2), we introduce a Schwartz function Ψ b on R n by setting c Ψ b = n times z }| {b ψ b ⊗ · · · ⊗ b ψ b . For j ∈ Z we can define the Littlewood-Paley operators corresponding to ψ and ψ b in the k th variable as the operators whose action on a function f on R n is as follows:∆ ψ,kj ( f )( x , . . . , x n ) = Z R f ( . . . , x k − y, . . . ) 2 j ψ (cid:0) j y (cid:1) dy and ∆ ψ b ,kj ( f )( x , . . . , x n ) = Z R f ( . . . , x k − y, . . . ) 2 j ψ b (cid:0) j y (cid:1) dy. Since b ψ b = 1 on the support of b ψ , c Ψ b (2 − j ξ , . . . , − j n ξ n ) = 1 on the support of b Ψ(2 − j ξ , . . . , − j n ξ n ) for each j , . . . , j n ∈ Z and so∆ ψ, j · · · ∆ ψ,nj n T σ ( f ) ( x , . . . , x n )= Z R n b f ( ξ , . . . ξ n ) b Ψ (cid:0) − j ξ , . . . , − j n ξ n (cid:1) σ ( ξ , . . . , ξ n ) e πi ( x ξ + ··· + x n ξ n ) dξ · · · dξ n = Z R n b f ( ξ , . . . ξ n ) c Ψ b (cid:0) − j ξ , . . . , − j n ξ n (cid:1) b Ψ (cid:0) − j ξ , . . . , − j n ξ n (cid:1) σ ( ξ , . . . , ξ n ) e πi ( x ξ + ··· + x n ξ n ) dξ · · · dξ n = Z R n (∆ ψ b , j · · · ∆ ψ b ,nj n f ) b ( ξ , . . . , ξ n ) b Ψ (cid:0) − j ξ , . . . , − j n ξ n (cid:1) σ ( ξ , . . . , ξ n ) e πi ( x ξ + ··· + x n ξ n ) dξ · · · dξ n ARCINKIEWICZ MULTIPLIER THEOREM 19 = Z R n j + ··· + j n (∆ ψ b , j · · · ∆ ψ b ,nj n f ) b (2 j ξ ′ , . . . , j n ξ ′ n ) b Ψ ( ξ ′ , . . . , ξ ′ n ) σ (2 j ξ ′ , . . . , j n ξ ′ ) e πi (2 j x ξ ′ + ··· +2 jn x n ξ ′ n ) dξ ′ · · · dξ ′ n = Z R n (∆ ψ b , j · · · ∆ ψ b ,nj n f )(2 − j y ′ , . . . , − j n y ′ n ) (cid:2) b Ψ D j ,...,j n σ (cid:3) b ( y ′ − j x , . . . , y ′ n − j n x n ) dy ′ · · · dy ′ n = Z R n (∆ ψ b , j · · · ∆ ψ b ,nj n f )(2 − j y + x , . . . , − j n y n + x n )[ b Ψ D j ,...,j n σ ] b ( y , . . . , y n ) dy · · · dy n = Z R n (cid:0) ∆ ψ b , j · · · ∆ ψ b ,nj n f (cid:1) (2 − j y + x , . . . , − j n y n + x n )(1 + | y | ) s · · · (1 + | y n | ) s n (1 + | y | ) s · · · (1 + | y n | ) s n [ b Ψ D j ,...,j n σ ] b ( y , . . . , y n ) dy · · · dy n . Applying H¨older’s inequality in the Lorentz-Marcinkiewicz setting (2.1), we obtainthat | ∆ ψ, j · · · ∆ ψ,nj n T σ ( f ) ( x , . . . , x n ) | is bounded by (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:0) ∆ ψ b , j · · · ∆ ψ b ,nj n f (cid:1) (2 − j y + x , . . . , − j n y n + x n )(1 + | y | ) s · · · (1 + | y n | ) s n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M ωs , − s d ( R n ,dy ··· dy n ) · (cid:13)(cid:13)(cid:13) (1 + | y | ) s · · · (1 + | y n | ) s n [ b Ψ D j ,...,j n σ ] b ( y , . . . , y n ) (cid:13)(cid:13)(cid:13) Λ ω − s ,s d ( R n ) . The first term in this product is estimated by Corollary 3.5 as follows: Since we areassuming 1 > s > /
2, there is a q such that 1 < /s < q <
2. Then for this q weget (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:0) ∆ ψ b , j · · · ∆ ψ b ,nj n f (cid:1) (2 − j y + x , . . . , − j n y n + x n )(1 + | y | ) s · · · (1 + | y n | ) s n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M ωs , − s d ( R n ) ≤ C M L q (cid:0) | ∆ ψ b , j · · · ∆ ψ b ,nj n f | (cid:1) ( x , . . . , x n ) . We estimate the second term in the product using Proposition 2.1 and Lemma 2.3,i.e., the Hausdorff-Young inequality adapted to these Lorentz spaces. We obtain (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n Y i =1 (1 + | y i | ) s i [ b Ψ D j ,...,j n σ ] b ( y , . . . , y n ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Λ ω − s ,s d ( R n ) ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n Y i =1 (1 + | y i | ) si [ b Ψ D j ,...,j n σ ] b ( y , . . . , y n ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Λ ω − s ,s d ( R n ) ≤ C (cid:13)(cid:13)(cid:13) Γ ( s , . . . , s n ) [ b Ψ D j ,...,j n σ ] (cid:13)(cid:13)(cid:13) Λ φs ,s d ( R n ) ≤ C e K. where we used the fact that 1 < /s <
2, which is a hypothesis of Lemma 2.3.We have now obtained the pointwise estimate(5.2) | ∆ ψ, j · · · ∆ ψ,nj n T σ ( f ) | ≤ C e K M L q (cid:0) | ∆ ψ b , j · · · ∆ ψ b ,nj n f | (cid:1) . Now let p ≥
2. Applying the product type Littlewood-Paley theorem, the Fefferman-Stein inequality, and estimate (5.2) we obtain k T σ ( f ) k L p ( R n ) ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X j ,...,j n ∈ Z (cid:12)(cid:12)(cid:12) ∆ ψ, j · · · ∆ ψ,nj n T σ ( f ) (cid:12)(cid:12)(cid:12) ! (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R n ) ≤ C e K (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X j ,...,j n ∈ Z (cid:12)(cid:12)(cid:12) M L q (cid:16) | ∆ ψ b , j · · · ∆ ψ b ,nj n f | (cid:17)(cid:12)(cid:12)(cid:12) ! (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R n ) ≤ C e K (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X j ,...,j n ∈ Z (cid:16) M (cid:16) | ∆ ψ b , j · · · ∆ ψ b ,nj n f | q (cid:17)(cid:17) q ! (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R n ) ≤ C e K (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X j ,...,j n ∈ Z (cid:16) M (cid:16) | ∆ ψ b , j · · · ∆ ψ b ,nj n f | q (cid:17)(cid:17) q ! q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) q L pq ( R n ) ≤ C ′ e K (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X j ,...,j n ∈ Z | ∆ ψ b , j · · · ∆ ψ b ,nj n f | q. q ! q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) q L pq ( R n ) ≤ C ′′ e K (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X j ,...,j n ∈ Z | ∆ ψ b , j · · · ∆ ψ b ,nj n f | ! (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R n ) ≤ C ′′′ e K k f k L p ( R n ) . The case 1 < p < (cid:3) Interpolation
In the previous section we have proved the main theorem under the extra assump-tion that 1 / < s <
1. This estimate will be useful for p near 1 or near ∞ while for p = 2 we can use the trivial L ∞ estimate for the multiplier. The final conclusion willbe a consequence of an interpolation result (Theorem 6.6) discussed in this section.We start with a few lemmas. In the proof we will use Boyd’s interpolation theorem(see [3, Theorem 1]) which states: If E is an r.i. space on R + such that 1 /p <α E ≤ β E < /p for some 1 < p < p < ∞ , then the r.i. space E (Ω) on (Ω , µ ) isinterpolation between L p ( µ ) and L p ( µ ), i.e., L p ( µ ) ∩ L p ( µ ) ֒ → E (Ω) ֒ → L p ( µ ) + L p ( µ ) and for any linear operator T on L p ( µ ) + L p ( µ ) such that T is bounded on L p j ( µ ) for j = 0 and j = 1, it follows that T is a bounded operator on E (Ω). Lemma 6.1.
Let Φ be a smooth function on R n with compactly supported Fouriertransform. Then, for any < s, s . . . , s n < , γ > and any function F on R n , wehave (cid:13)(cid:13)(cid:13) Γ ( s, s , . . . , s n ) (cid:2)b Φ F (cid:3)(cid:13)(cid:13)(cid:13) Λ φs,γ ( R n ) ≤ k Γ ( s, s , . . . , s n ) F k Λ φs,γ ( R n ) . Lemma 6.2.
Let < s < and γ > . Then, for any t , . . . , t n ∈ R , we have k Γ ( it , . . . , it n ) f k Λ φs,γ ( R n ) ≤ C ( p, n )(1 + | t | ) · · · (1 + | t n | ) k f k Λ φs,γ ( R n ) . ARCINKIEWICZ MULTIPLIER THEOREM 21
Lemma 6.3.
Let < s < . Let m be a function satisfying (1.1) . Then, for any γ > , we have k T m ( f ) k Λ φs,γ ( R n ) ≤ C ( p, n ) k f k Λ φs,γ ( R n ) . All these lemmas can be proved in the following way.
Proof.
It is easy to check that if ϕ ∈ P and Λ ϕ is the Lorentz space on R + , then k σ t k E = m ϕ ( t ) for all t >
0. This implies that α Λ ϕ = γ ϕ and β Λ ϕ = δ ϕ .Now we choose p and p ∈ (1 , ∞ ) such that 1 /p < s < /p and let E := Λ φ s,γ .Combining the above fact with Proposition 2.1, we conclude that1 /p < α E = β E = s < /p . Since the estimates hold for L p and L p in place of the Lorentz space, Boyd’sinterpolation theorem completes the proof. (cid:3) We will use the following lemma (see [14, Lemma 2.1]).
Lemma 6.4.
Let < p ≤ p < ∞ and define p via /p = (1 − θ ) /p + θ/p ,where < θ < . Given f ∈ C ∞ ( R n ) and ε > , there exist smooth functions h εj , j = 1 , . . . , N ε , supported in cubes with disjoint interiors, and there exist nonzerocomplex constants c εj such that the functions (6.1) f εz = N ε X j =1 | c εj | pp (1 − z )+ pp z h εj satisfy (6.2) (cid:13)(cid:13) f εθ − f (cid:13)(cid:13) L p + (cid:13)(cid:13) f εθ − f (cid:13)(cid:13) L p + (cid:13)(cid:13) f εθ − f (cid:13)(cid:13) L < ε and (6.3) k f εit k p L p ≤ k f k pL p + ε ′ , k f ε it k p L p ≤ k f k pL p + ε ′ , where ε ′ depends on ε, p, k f k L p and tends to zero as ε → . The next lemma is a variant of Lemma 3.7 from [16].
Lemma 6.5.
Let < α, β < , γ > . Then for some constant C ( α, β, γ ) we have (6.4) Z ∞ ( f ∗ ( r ) r β − α ) ∗ ( y ) φ α,γ ( y ) dyy ≤ C ( α, β, γ ) Z ∞ f ∗ ( r ) φ β,γ ( r ) drr . Proof.
Recall that for given s ∈ (0 ,
1) and γ >
0, we have the equivalence φ s,γ ( t ) ≈ t s log γ (cid:0) e + t (cid:1) on (0 , ∞ ). Now observe that the estimate (6.4) is trivial when β ≤ α as ( f ∗ ( r ) r β − α ) ∗ = f ∗ ( r ) r β − α . Thus we may assume that β > α in the proof below.We may also assume that Z ∞ f ∗ ( r ) r β − log γ (cid:16) e + 1 r (cid:17) dr < ∞ , otherwise the right-hand side of (6.4) is infinite. Thensup r> f ∗ ( r ) r β log γ (cid:16) e + 1 r (cid:17) ≤ C, and thus lim r →∞ f ∗ ( r ) r β − α = 0. Since the set of discontinuity points of f ∗ is atmost countable ( f ∗ is right continuous), we may assume without loss of generality that function f ∗ is continuous. Then sup y ≤ r< ∞ f ∗ ( r ) r β − α is attained for any y > M = { y ∈ (0 , ∞ ) : sup y ≤ r< ∞ f ∗ ( r ) r β − α > f ∗ ( y ) y β − α } is open. Hence, M is a countable union of open intervals, namely, M = S k ∈ S ( a k , b k ),where S is a countable set of positive integers. Also, observe that if y ∈ ( a k , b k ),then sup y ≤ r< ∞ f ∗ ( r ) r β − α = f ∗ ( b k ) b β − αk . We have Z ∞ ( f ∗ ( r ) r β − α ) ∗ ( y ) y α − log γ (cid:16) e + 1 y (cid:17) dy ≤ Z ∞ sup y ≤ r< ∞ f ∗ ( r ) r β − α y α − log γ (cid:16) e + 1 y (cid:17) dy ≤ Z (0 , ∞ ) \ M f ∗ ( y ) y β − log γ (cid:16) e + 1 y (cid:17) dy + X k ∈ S f ∗ ( b k ) b β − αk Z b k a k y α − log γ (cid:16) e + 1 y (cid:17) dy . Furthermore, for every k ∈ S , f ∗ ( b k ) b β − αk Z b k a k y α − log γ (cid:16) e + 1 y (cid:17) dy ≤ f ∗ ( b k ) b β − αk Z b k max( a k , bk ) y α − log γ (cid:16) e + 1 y (cid:17) dy · R b k y α − log γ ( e + y ) dy R b kbk y α − log γ ( e + y ) dy = C α,γ f ∗ ( b k ) b β − αk Z b k max( a k , bk ) y α − log γ (cid:16) e + 1 y (cid:17) dy ≤ C α,β,γ Z b k a k f ∗ ( y ) y β − log γ (cid:16) e + 1 y (cid:17) dy . Therefore, Z ∞ ( f ∗ ( r ) r β − α ) ∗ ( y ) y α − log γ (cid:16) e + 1 y (cid:17) dy ≤ Z ∞ f ∗ ( y ) y β − log γ (cid:16) e + 1 y (cid:17) dy + C α,β,γ X k ∈ S Z b k a k f ∗ ( y ) y β − log γ (cid:16) e + 1 y (cid:17) dy ≤ ( C α,β,γ + 1) Z ∞ f ∗ ( y ) y β − log γ (cid:16) e + 1 y (cid:17) dy . This proves (6.4). (cid:3)
The main interpolation tool in this work is the following.
Theorem 6.6.
Let < p < ∞ and suppose that < s ≤ s ≤ · · · ≤ s n < andthat < s ≤ s ≤ · · · ≤ s n < . Assume that exactly d of the numbers s , . . . , s n ARCINKIEWICZ MULTIPLIER THEOREM 23 are equal to s , and exactly d of the numbers s , . . . , s n are equal to s . Let Ψ be asin (1.2) . Suppose that for all nonzero f ∈ C ∞ ( R n ) we have (6.5) (cid:13)(cid:13) T σ f (cid:13)(cid:13) L p ( R n ) ≤ K sup j ,...,j n ∈ Z (cid:13)(cid:13) Γ (cid:0) s , . . . , s n (cid:1) [ b Ψ D j ,...,j n σ ] (cid:13)(cid:13) Λ φs ,s d ( R n ) k f k L p ( R n ) and (6.6) (cid:13)(cid:13) T σ f (cid:13)(cid:13) L p ( R n ) ≤ K sup j ,...,j n ∈ Z (cid:13)(cid:13) Γ (cid:0) s , . . . , s n (cid:1) [ b Ψ D j ,...,j n σ ] (cid:13)(cid:13) Λ φs , (1 − s d ( R n ) k f k L p ( R n ) . Let < θ < and suppose p = 1 − θp + θ , s j = (1 − θ ) s j + θs j , j = 1 , . . . , n. Then there is a constant C ∗ = C ∗ ( p , θ, n, d, ψ, s j , s j ) such that for all f ∈ C ∞ ( R n ) (cid:13)(cid:13) T σ f (cid:13)(cid:13) L p ( R n ) ≤ C ∗ K − θ K θ sup j ,...,j n ∈ Z (cid:13)(cid:13) Γ ( s , . . . , s n ) [ b Ψ D j ,...,j n σ ] (cid:13)(cid:13) Λ φs ,d ( R n ) k f k L p ( R n ) . Proof.
Let us fix a function σ such that(6.7) sup j ,...,j n ∈ Z (cid:13)(cid:13)(cid:13) Γ( s , . . . , s n ) (cid:2) b Ψ D j ,...,j n σ (cid:3)(cid:13)(cid:13)(cid:13) Λ φs ,d ( R n ) < ∞ and for j , . . . , j n ∈ Z define ϕ j ,...,j n = Γ( s , . . . , s n ) (cid:2) b Ψ D j ,...,j n σ (cid:3) . Since ϕ j ,...,j n ∈ Λ φ s ,d ( R n ), we have sup λ> φ s ,d ( λ ) ϕ ∗ j ,...,j n ( λ ) < ∞ and so ϕ ∗ j ,...,j n ( λ )converges to 0 as λ → ∞ . Now by [2, Corollary 7.6 in Chapter 2], there is a measurepreserving transformation h j ,...,j n : R n → (0 , ∞ ) such that(6.8) | ϕ j ,...,j n | = ϕ ∗ j ,...,j n ◦ h j ,...,j n . Recall that s ≤ · · · ≤ s n and s ≤ · · · ≤ s n . For z ∈ C with 0 ≤ Re( z ) ≤
1, wedefine complex polynomials P ρ ( z ) = s ρ (1 − z ) + s ρ z for ρ = 1 , , . . . , n . Let c Ψ b = b ψ b ⊗ · · · ⊗ b ψ b where ψ b is defined in (5.1). We definethe family of multipliers(6.9) σ z = X k ,...,k n ∈ Z D − k ,..., − k n (cid:20)c Ψ b Γ (cid:0) − P ( z ) , . . . , − P n ( z ) (cid:1)h ϕ k ,...,k n h s − P ( z ) k ,...,k n i(cid:21) . As P j ( θ ) = s j for 1 ≤ j ≤ n and P k ,...,k n ∈ Z b Ψ (cid:0) − k ξ , . . . , − k n ξ n (cid:1) = 1 when all ξ k = 0, it follows that σ θ = σ a.e.Fix f, g ∈ C ∞ ( R n ). Given ǫ > f ǫz and g ǫz as in Lemma 6.4. Thus we have k f ǫθ − f k L p + k f ǫθ − f k L < ǫ , k g ǫθ − g k L p ′ + k g ǫθ − g k L ≤ ǫ , k f ǫit k p L p ( R n ) ≤ k f k pL p ( R n ) + ǫ ′ , (cid:13)(cid:13) f ǫ it (cid:13)(cid:13) L ( R n ) ≤ k f k pL p ( R n ) + ǫ ′ , k g ǫit k p ′ L p ′ ( R n ) ≤ k g k p ′ L p ′ ( R n ) + ǫ ′ , (cid:13)(cid:13) g ǫ it (cid:13)(cid:13) L ( R n ) ≤ k g k p ′ L p ′ ( R n ) + ǫ ′ . Now define on the unit strip { z ∈ C : 0 ≤ Re( z ) ≤ } the following function(6.10) F ( z ) = Z R n σ z ( ξ ) b f ǫz ( ξ ) b g ǫz ( ξ ) dξ = Z R n T σ z ( f ǫz )( x ) g ǫz ( − x ) dx which is analytic in the interior of this strip and is continuous on its closure. H¨older’sinequality and one hypothesis of the theorem give | F ( it ) | ≤ k T σ it ( f ǫit ) k L p k g ǫit k L p ′ ≤ K sup j ,...,j n ∈ Z (cid:13)(cid:13)(cid:13) Γ (cid:0) s , . . . , s n (cid:1) (cid:2) b Ψ D j ,...,j n σ it (cid:3)(cid:13)(cid:13)(cid:13) Λ φs ,s d k f ǫit k L p k g ǫit k L p ′ . (6.11)Using the definition of σ z with z = it , we have b Ψ D j ,...,j n σ it = X k ,...,k n ∈ Z b Ψ D j − k ,...,j n − k n "c Ψ b Γ (cid:0) − P ( it ) , . . . , − P n ( it ) (cid:1)h ϕ k ,...,k n h s − P ( it ) k ,...,k n i . In view of the support properties of the bumps b Ψ and c Ψ b , all terms in the sum aboveare zero if k i / ∈ { j i − , j i − , j i , j i + 1 , j i + 2 } for some i ∈ { , . . . , n } . Using thisobservation and Lemma 6.1 with b Φ = b Ψ D a ,...,a n c Ψ b , we write (cid:13)(cid:13)(cid:13) Γ (cid:0) s , . . . , s n (cid:1) (cid:2) b Ψ D j ,...,j n σ it (cid:3)(cid:13)(cid:13)(cid:13) Λ φs ,s d ≤ X ≤ i ≤ n − ≤ a i ≤ (cid:13)(cid:13)(cid:13)(cid:13) Γ (cid:0) s , . . . , s n (cid:1) (cid:20) b Ψ (cid:0) D a ,...,a n c Ψ b (cid:1) D a ,...,a n n Γ (cid:0) − P ( it ) , . . . , − P n ( it ) (cid:1)(cid:0) ϕ j + a ,...,j n + a n h s − P ( it ) j + a ,...,j n + a n (cid:1)o(cid:21)(cid:13)(cid:13)(cid:13)(cid:13) Λ φs ,s d ≤ X ≤ i ≤ n − ≤ a i ≤ (cid:13)(cid:13)(cid:13)(cid:13) Γ (cid:0) s , . . . , s n (cid:1) D a ,...,a n n Γ (cid:0) − s , . . . , − s n (cid:1) Γ (cid:0) it ( s − s ) , . . . , it ( s n − s n ) (cid:1)(cid:0) ϕ j + a ,...,j n + a n h s − P ( it ) j + a ,...,j n + a n (cid:1)o(cid:13)(cid:13)(cid:13)(cid:13) Λ φs ,s d ≤ C X ≤ i ≤ n − ≤ a i ≤ (cid:13)(cid:13)(cid:13)(cid:13) Γ (cid:0) it ( s − s ) , . . . , it ( s n − s n ) (cid:1) h ϕ j + a ,...,j n + a n h s − P ( it ) j + a ,...,j n + a n i (cid:13)(cid:13)(cid:13)(cid:13) Λ φs ,s d , as − P j ( it ) = − s j + it ( s j − s j ). In the last inequality we made use of the fact thatthe function n Y i =1 (cid:18) π | ξ i | π | ξ i / a i | (cid:19) s i / satisfies (1.1) and thus Lemma 6.3 applies. We continue estimating as follows: C X ≤ i ≤ n − ≤ a i ≤ (cid:13)(cid:13)(cid:13)(cid:13) Γ (cid:0) it ( s − s ) , . . . , it ( s n − s n ) (cid:1) h ϕ j + a ,...,j n + a n h s − P ( it ) j + a ,...,j n + a n i (cid:13)(cid:13)(cid:13)(cid:13) Λ φs ,s d ( R n ) ≤ C X ≤ i ≤ n − ≤ a i ≤ (1 + | t | ) n (cid:13)(cid:13)(cid:13) ϕ j + a ,...,j n + a n h s − s j + a ,...,j n + a n (cid:13)(cid:13)(cid:13) Λ φs ,s d ( R n ) ARCINKIEWICZ MULTIPLIER THEOREM 25 = C (1 + | t | ) n X ≤ i ≤ n − ≤ a i ≤ (cid:13)(cid:13)(cid:13) ϕ ∗ j + a ,...,j n + a n ( r ) r s − s (cid:13)(cid:13)(cid:13) Λ φs ,s d ((0 , ∞ ) ,dr ) ≤ C (1 + | t | ) n X ≤ i ≤ n − ≤ a i ≤ (cid:13)(cid:13) ϕ ∗ j + a ,...,j n + a n (cid:13)(cid:13) Λ φs ,s d (0 , ∞ ) = C (1 + | t | ) n X ≤ i ≤ n − ≤ a i ≤ k ϕ j + a ,...,j n + a n k Λ φs ,s d ( R n ) ≤ C (1 + | t | ) n sup j ,...,j n ∈ Z k ϕ j ,...,j n k Λ φs ,d ( R n ) , where we used successively Lemma 6.2, the fact that Re P ( it ) = s , identity (6.8)together with the fact that h j ,...,j n is measure-preserving, and Lemma 6.5. Insertingthis estimate in (6.11) and using Lemma 6.4 we obtain | F ( it ) | ≤ CK (1 + | t | ) n sup j ,...,j n ∈ Z k ϕ j ,...,j n k Λ φs ,d ( R n ) (cid:0) k f k pL p + ǫ ′ (cid:1) p (cid:0) k g k p ′ L p ′ + ǫ ′ (cid:1) p ′ . A similar argument using the inequality k ϕ j ,...,j n k Λ φs , (1 − s d ( R n ) ≤ k ϕ j ,...,j n k Λ φs ,d ( R n ) yields | F (1 + it ) | ≤ CK (1 + | t | ) n sup j ,...,j n ∈ Z k ϕ j ,...,j n k Λ φs ,d ( R n ) (cid:0) k f k pL p + ǫ ′ (cid:1) (cid:0) k g k p ′ L p ′ + ǫ ′ (cid:1) . Moreover, for τ ∈ [0 , | F ( τ + it ) | ≤ A τ ( t ) where A τ ( t ) has at mostpolynomial growth as | t | → ∞ ; we prove this assertion at the end. Thus we canapply Hirschman’s lemma ([11, Lemma 1.3.8]). Using the estimates for | F ( it ) | and | F (1 + it ) | , for θ ∈ (0 , | F ( θ ) | ≤ C ∗ K − θ K θ sup j ,...,j n ∈ Z k ϕ j ,...,j n k Λ φs ,d ( R n ) (cid:0) k f k pL p + ǫ ′ (cid:1) p (cid:0) k g k p ′ L p ′ + ǫ ′ (cid:1) p ′ . We write (cid:12)(cid:12)(cid:12)(cid:12) F ( θ ) − Z R n \ T σ ( f ) b g ( ξ ) dξ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z R n σ ( ξ ) b f ǫθ ( ξ ) b g ǫθ ( ξ ) dξ − Z R n σ ( ξ ) b f ( ξ ) b g ( ξ ) dξ (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13) σ k L ∞ h(cid:13)(cid:13) f ǫθ − f (cid:13)(cid:13) L (cid:13)(cid:13) g (cid:13)(cid:13) L + (cid:13)(cid:13) g ǫθ − g (cid:13)(cid:13) L (cid:13)(cid:13) f (cid:13)(cid:13) L i , which tends to zero as ǫ → ǫ ′ → (cid:12)(cid:12)(cid:12)(cid:12) Z R n \ T σ ( f ) b g ( ξ ) dξ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ∗ K − θ K θ sup j ,...,j n ∈ Z k ϕ j ,...,j n k Λ φs ,d ( R n ) k f k L p k g k L p ′ . But the integral on the left is equal to R R n T σ ( f )( x ) g ( − x ) dx . Taking the supremumover all functions g ∈ C ∞ ( R n ) with k g k L p ′ ≤ f ∈ C ∞ ( R n ): k T σ ( f ) k L p ( R n ) ≤ C ∗ K − θ K θ sup j ,...,j n ∈ Z k ϕ j ,...,j n k Λ φs ,d ( R n ) k f k L p ( R n ) . Notice that the constant C ∗ depends on the parameters indicated in the statement.We now return to the assertion that | F ( τ + it ) | ≤ A τ ( t ), where A τ ( t ) has at mostpolynomial growth in | t | , which was one of the hypotheses in Hirschman’s lemma.Let z = τ + it where t ∈ R and 0 ≤ τ ≤
1. We use that | F ( τ + it ) | ≤ k σ τ + it k L ∞ k f τ + it k L k g τ + it k L , and we notice that in view of (6.1), the L norms of f τ + it and g τ + it are boundedby constants independent of t . We now estimate k σ z k L ∞ . Let E be the set of all( ξ , . . . , ξ n ) ∈ R n with some ξ i = 0. Then for all ( ξ , . . . , ξ n ) ∈ R n \ E there areonly finitely many indices k i in the summation defining σ z ( ξ , . . . , ξ n ) that producea nonzero term, in fact the indices with | ξ i | / ≤ k i ≤ | ξ i | for all i ∈ { , . . . , n } .Also, P ρ ( τ + it ) = P ρ ( τ ) + ( s − s )( it ), which implies thatΓ (cid:0) − P ( τ + it ) , . . . , − P n ( τ + it ) (cid:1) = Γ (cid:0) − P ( τ ) , . . . , − P n ( τ ) (cid:1) Γ (cid:0) it ( s − s ) , . . . , it ( s n − s n ) (cid:1) . (6.12)Applying identity (6.12), and using successively Proposition 4.1, Lemma 6.2, thefact that Re P ( τ + it ) = P ( τ ), identity (6.8) together with the fact that h k ,...,k n ismeasure-preserving, and Lemma 6.5, we estimate k σ τ + it k L ∞ bysup ξ ∈ R n \ E X ≤ i ≤ n | ξi | ≤ ki ≤ | ξ i | (cid:13)(cid:13)(cid:13)(cid:13) Γ (cid:0) − P ( τ + it ) , . . . , − P n ( τ + it ) (cid:1) h ϕ k ,...,k n h s − P ( τ + it ) k ,...,k n i (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ≤ C (1 + | t | ) n sup ξ ∈ R n \ E X ≤ i ≤ n | ξi | ≤ ki ≤ | ξ i | (cid:13)(cid:13)(cid:13) ϕ k ,...,k n h s − P ( τ ) k ,...,k n (cid:13)(cid:13)(cid:13) Λ φP τ ) , (1 − P τ )) d ( R n ) ≤ C (1 + | t | ) n sup ξ ∈ R n \ E X ≤ i ≤ n | ξi | ≤ ki ≤ | ξ i | (cid:13)(cid:13) ϕ ∗ k ,...,k n ( r ) r s − P ( τ ) (cid:13)(cid:13) Λ φP τ ) , (1 − P τ )) d (0 , ∞ ) ≤ C (1 + | t | ) n sup ξ ∈ R n \ E X ≤ i ≤ n | ξi | ≤ ki ≤ | ξ i | (cid:13)(cid:13) ϕ ∗ k ,...,k n (cid:13)(cid:13) Λ φs , (1 − P τ )) d (0 , ∞ ) ≤ C (1 + | t | ) n n sup k ,...,k n ∈ Z k ϕ k ,...,k n k Λ φs ,d ( R n ) , and the last expression is finite in view of assumption (6.7). This proves that | F ( τ + it ) | ≤ A τ ( t ), where A τ ( t ) ≤ C ′ (1 + | t | ) n . (cid:3) To prove Theorem 1.1 we apply Theorem 6.6 as follows: For the given p with1 < p < p = 1 + ǫ for some small number ǫ , and we define θ in terms of(1 − θ ) /p + θ/ /p .Given 0 < s ≤ · · · ≤ s n with exactly d numbers among s , . . . , s n equal to s ,pick < s ≤ · · · ≤ s n and 0 < s ≤ · · · ≤ s n ≤ / s j = (1 − θ ) s j + θs j .This relationship maintains proportions, and as the sequences are all increasing, itmust be the case that the first d + 1 terms in each sequence are equal. We pick thesesequences so that s = · · · = s d +1 = + ε and s = · · · = s d +1 . We note that s canbe found thanks to the assumption s > /p − /
2. Inequality (6.5) follows from thespecial case 1 / < s < Remark 6.7.
Assume that all assumptions of Theorem 1.1 are satisfied and, inaddition, s ≤ /
2. Let δ >
0. We claim that inequality (1.6) holds with the(smaller) constant K = sup j ,...,j n ∈ Z (cid:13)(cid:13)(cid:13) Γ ( s , . . . , s n ) (cid:2) b Ψ D j ,...,j n σ (cid:3)(cid:13)(cid:13)(cid:13) Λ φs , (1 − s δ ) d ( R n ) . ARCINKIEWICZ MULTIPLIER THEOREM 27
This can be proved by employing a slight modification of the proof of Theorem 6.6.Namely, we replace equation (6.9) by σ z = X k ,...,k n ∈ Z D − k ,..., − k n (cid:20)c Ψ b Γ (cid:0) − P ( z ) , . . . , − P n ( z ) (cid:1)h ϕ k ,...,k n h s − P ( z ) k ,...,k n (log( e + h − k ,...,k n )) ( P ( z ) − s ) d i(cid:21) and define the function F by (6.10). Then one can show that | F ( it ) | ≤ CK (1 + | t | ) n sup j ,...,j n ∈ Z k ϕ j ,...,j n k Λ φs ,d (2 s − s ( R n ) (cid:0) k f k pL p + ǫ ′ (cid:1) p (cid:0) k g k p ′ L p ′ + ǫ ′ (cid:1) p ′ and | F (1+ it ) | ≤ CK (1+ | t | ) n sup j ,...,j n ∈ Z k ϕ j ,...,j n k Λ φs ,d (1 − s ( R n ) (cid:0) k f k pL p + ǫ ′ (cid:1) (cid:0) k g k p ′ L p ′ + ǫ ′ (cid:1) . This then implies k T σ ( f ) k L p ( R n ) ≤ C ∗ K − θ K θ sup j ,...,j n ∈ Z k ϕ j ,...,j n k Λ φs ,d (2 s − s ( R n ) k f k L p ( R n ) . Choosing all parameters as in the proof of Theorem 1 . ǫ < δ/ s > / . K = sup j ,...,j n ∈ Z (cid:13)(cid:13)(cid:13) Γ ( s , . . . , s n ) (cid:2) b Ψ D j ,...,j n σ (cid:3)(cid:13)(cid:13)(cid:13) Λ φs ,s d ( R n ) ;this was proved in Section 5. References [1] J. J. Benedetto and H. P. Heinig,
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