A family of Hardy type spaces on nondoubling manifolds
aa r X i v : . [ m a t h . F A ] A ug A FAMILY OF HARDY TYPE SPACESON NONDOUBLING MANIFOLDS
ALESSIO MARTINI, STEFANO MEDA, AND MARIA VALLARINO
Abstract.
We introduce a decreasing one-parameter family X γ ( M ), γ > h ( M ) on certain nondou-bling Riemannian manifolds with bounded geometry and we investigate theirproperties. In particular, we prove that X / ( M ) agrees with the space of allfunctions in h ( M ) whose Riesz transform is in L ( M ), and we obtain thesurprising result that this space does not admit an atomic decomposition. Introduction
In their seminal paper [FS] C. Fefferman and E.M. Stein defined the classicalHardy space H ( R n ) as follows: H ( R n ) := { f ∈ L ( R n ) : (cid:12)(cid:12) ∇ ( − ∆) − / f (cid:12)(cid:12) ∈ L ( R n ) } ; (1.1)here ∇ and ∆ denote the Euclidean gradient and Laplacian, respectively. Follow-ing up earlier work of D.L. Burkholder, R.F. Gundy and M.L. Silverstein [BGS],Fefferman and Stein obtained several characterisations of H ( R n ) in terms of var-ious maximal operators and square functions, thereby starting the real variabletheory of Hardy spaces. Their analysis was complemented by R.R. Coifman [Coi],who showed that H ( R ) admits an atomic decomposition. This result was laterextended to higher dimensions by R. Latter [La].It is natural to speculate whether an analogue of the results of Fefferman–Stein,Coifman and Latter holds in different settings. In other words, one may ask whatis the most appropriate way to define Hardy spaces in settings other than R n andwhether different definitions lead to the same spaces. In this paper we will considerthis problem on a class of nondoubling Riemannian manifolds.There is a huge literature concerning this question on manifolds or on even moreabstract sorts of spaces and it is virtually impossible to give an account of the mainresults in the field. Thus, without any pretence of exhaustiveness, we mention justa few contributions, which we consider the most relevant to our discussion. It is fairto say that most of the results in the literature are concerned with settings wherethe relevant metric and measure satisfy the doubling condition; while these worksdo not directly apply to the manifolds considered here, they nevertheless play aparadigmatic role in the development of the subject and it would be impossible toleave them out of our discussion.In the context of spaces of homogeneous type, Coifman and G. Weiss [CW] de-fined an atomic Hardy space which generalises the Euclidean one. Various maximalfunction characterisations of such Hardy space have been obtained by a number of Mathematics Subject Classification.
Key words and phrases.
Hardy space, atom, noncompact manifold, exponential growth, Riesztransform.Work partially supported by PRIN 2015 “Real and complex manifolds: geometry, topologyand harmonic analysis” and by the EPSRC Grant EP/P002447/1. The authors are members ofthe Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Applicazioni (GNAMPA)of the Istituto Nazionale di Alta Matematica (INdAM) . authors under additional assumptions on the underlying metric (see, e.g., [U1, YZ1]and references therein); however, a characterisation in terms of singular integrals,similar to the Euclidean one via Riesz transforms, remains in general a deceptiveproblem. A complete characterisation of the Coifman–Weiss Hardy space in termsof maximal functions and Riesz transforms was carried over by G. Folland and Stein[FS] and by M. Christ and D. Geller [CG] in the case of stratified groups, followinga deep result of A. Uchiyama [U2]. Further results in this direction were obtainedby J. Dziuba´nski and K. Jotsaroop [DJ] in R n , but with the Grushin operator play-ing the role of the Euclidean Laplacian, and by Dziuba´nski and J. Zienkiewicz (see[DZ1, DZ2] and the references therein) for Schr¨odinger operators with nonnegativepotentials satisfying certain additional assumptions.Within the class of Riemannian manifolds with doubling Riemannian measure,a consequence of works of various authors [AMR, HLMMY, DKKP] (see also thereferences therein) is that, under mild geometric conditions, the Hardy spaces de-fined in terms of the heat maximal operator and the Poisson maximal operatoragree, and coincide with an atomic Hardy space defined in terms of appropriateatoms (these are atoms naturally associated to the Laplace–Beltrami operator andmay differ considerably from those defined by Coifman and Weiss). Furthermore,in this setting it is known that the Riesz–Hardy space contains the atomic space,but, to the best of our knowledge, the question whether this inclusion is proper isstill open.In this paper we consider Riemannian manifolds M with positive injectivity ra-dius, Ricci tensor bounded from below and spectral gap. Notice that M , equippedwith the Riemannian distance, is not a space of homogeneous type in the senseof Coifman and Weiss, for the doubling condition fails for large balls. The corre-sponding theory of atomic Hardy type spaces has been developed only quite recently[Io, CMM1, T, MMV3], and it differs remarkably from that of H ( R n ). A relatednondoubling setting where interesting results concerning aspects of this programmehave been developed has been considered in [V, MOV].The final outcome of our research, which will be described in detail in the presentpaper and in other forthcoming papers, is that different definitions of Hardy spaces(atomic, via Riesz transform, via maximal operators) on certain manifolds with ex-ponential volume growth may very well lead to different spaces. Related interestingpartial results are [A, Corollary 6.3] and [Lo].Our work is inspired by a series of papers [MMV2, MMV3], by the Ph.D. thesis[Vo] and by the recent work [CM] on graphs. Specifically, in [MMV2] the Au-thors introduced a sequence X k ( M ) of strictly decreasing Banach spaces, whichare isometric copies of the Hardy type space H ( M ), introduced by A. Carbonaro,G. Mauceri and Meda in [CMM1]. This space differs from the classical Hardy spaceof Coifman–Weiss [CW].Volpi [Vo] modified this construction by letting the Hardy–Goldberg type space h ( M ), introduced by M. Taylor in [T] and further generalised by Meda andVolpi [MVo], play the role of the space H ( M ) of Carbonaro, Mauceri and Meda.The resulting sequence of spaces is named X k ( M ), instead of X k ( M ). Of course X k ( M ) ⊇ X k ( M ), for h ( M ) properly contains H ( M ). It may be worth recall-ing that h ( M ) is the analogue on M of the classical space h ( R n ) introduced byD. Goldberg in [Go] and further investigated on specific measure metric spaces invarious papers, including [HMY, YZ1, YZ2, BDL] (see also the references therein).In this paper we take a step further, and consider a one-parameter family ofspaces X γ ( M ), where γ is a positive real number, which agree with those intro-duced in [Vo] when γ is a positive integer. Specifically, the space X γ ( M ) is just U γ (cid:2) h ( M ) (cid:3) , where U = L ( I + L ) − and L is the (positive) Laplace–Beltrami FAMILY OF HARDY TYPE SPACES 3 operator on M . It is not hard to see that U is injective on L ( M ), hence so is U γ .The space X γ ( M ) is endowed with the norm that makes U γ an isometry between h ( M ) and X γ ( M ), i.e., (cid:13)(cid:13) f (cid:13)(cid:13) X γ ( M ) := (cid:13)(cid:13) U − γ f (cid:13)(cid:13) h ( M ) .The idea of considering noninteger values of γ is taken from [CM], where ananalogue of X γ ( M ) is defined on certain graphs with exponential volume growth.However, the case of Riemannian manifolds we consider here requires substantialrefinements of the theory developed in [CM].The spaces X γ ( M ) play a central role in our analysis of Hardy type spaces. Weprove that X γ ( M ) is a decreasing family of Banach spaces, each of which interpo-lates with L ( M ). We also show that the imaginary powers of the Laplace–Beltramioperator L are bounded from X γ ( M ) to h ( M ) for all γ >
0, thus providing an end-point counterpart to their L p boundedness for p ∈ (1 , ∞ ); notice that the imaginarypowers of L may not be bounded from h ( M ) to L ( M ) [MMV4].We also prove that X γ ( M ) does not admit an atomic decomposition when γ isnot an integer, at least in the case of symmetric spaces of the noncompact typeand real rank one. More precisely, we show that the space of compactly supportedelements of X γ ( M ) is not dense in X γ ( M ).The extension to noninteger values of the parameter γ is a posteriori motivatedby one of our main results, which states that X / ( M ) = { f ∈ h ( M ) : | R f | ∈ L ( M ) } . (1.2)Here R denotes the Riesz transform ∇ L − / on M , and ∇ is the Riemanniangradient. Notice that we do not prove here that the Riesz–Hardy space H R ( M ),defined by H R ( M ) := { f ∈ L ( M ) : | R f | ∈ L ( M ) } , (1.3)agrees with X / ( M ). The proof of this equivalence requires (1.2) together withmore sophisticated real variable methods, and will be given in [MVe]. In conjunctionwith the results of the present paper, this equivalence implies the perhaps surprisingresult that H R ( M ) does not admit in general an atomic decomposition.The relations between the spaces X γ ( M ) for different values of γ > H H ( M ) and H P ( M ) defined in terms of the heat and the Poissonmaximal operators will be discussed in detail in [MaMV], yielding another possiblysurprising result: the spaces H R ( M ), H H ( M ) and H P ( M ) may all differ in thiscontext.We shall use the “variable constant convention”, and denote by C , possibly withsub- or superscripts, a constant that may vary from place to place and may dependon any factor quantified (implicitly or explicitly) before its occurrence, but not onfactors quantified afterwards.2. Background on Hardy type spaces
Let M denote a connected, complete n -dimensional Riemannian manifold ofinfinite volume with Riemannian measure µ . Denote by L the positive Laplace–Beltrami operator on M , by b the bottom of the L ( M ) spectrum of L , and set β = lim sup r →∞ (cid:2) log µ (cid:0) B r ( o ) (cid:1)(cid:3) / (2 r ), where o is any reference point of M and B r ( o )denotes the ball centred at o of radius r . By a result of Brooks, b ≤ β [Br].We denote by B the family of all geodesic balls on M . For each B in B wedenote by c B and r B the centre and the radius of B respectively. Furthermore, foreach positive number λ , we denote by λ B the ball with centre c B and radius λ r B .For each scale parameter s in R + , we denote by B s the family of all balls B in B such that r B ≤ s . A. MARTINI, S. MEDA, AND M. VALLARINO
In this paper we make the following assumptions on the geometry of themanifold :(i) the injectivity radius of M is positive;(ii) the Ricci tensor is bounded from below;(iii) M has spectral gap, to wit b > M satisfies the localdoubling condition and supports a local scaled L -Poincar´e inequality, which areimplied by (ii) above, but they do not require (i) and (iii). We believe that it is notworth keeping track of the minimal assumptions under which each of the resultsbelow holds, and assume throughout that M satisfies (i)–(iii) above.It is well known that for manifolds satisfying (i)–(iii) above the following prop-erties hold:(a) there are positive constants α and C such that µ ( B ) ≤ C r αB e β r B ∀ B ∈ B \ B , (2.1)where β is the constant defined at the beginning of this section;(b) (see [MMV3, Remark 2.3]) there exists a positive constant C such that C − r nB ≤ µ ( B ) ≤ C r nB ∀ B ∈ B ; (2.2)(c) as a consequence of (a) and (b) the measure µ is locally doubling , i.e., forevery s > D s such that µ (2 B ) ≤ D s µ ( B ) ∀ B ∈ B s ;(d) M possesses a local scaled L -Poincar´e inequality , i.e., for each R < ∞ there exists a constant C , depending on R , such that Z B (cid:12)(cid:12) f − f B (cid:12)(cid:12) d µ ≤ C r B Z B (cid:12)(cid:12) ∇ f (cid:12)(cid:12) d µ , (2.3)for all balls B in B R ;(e) the heat semigroup { H t } is ultracontractive , in the sense that H t := e − t L maps L ( M ) into L ( M ) and satisfies the following estimate [Gr, Section7.5]: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C e − bt t − n/ (1 + t ) n ∀ t ∈ R + ;it follows by interpolation that, for every p ∈ (1 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ C [e − bt t − n/ (1 + t ) n ] /p ′ ∀ t ∈ R + , (2.4)and in particular lim t →∞ (cid:13)(cid:13) H t f (cid:13)(cid:13) L p = 0 ∀ f ∈ L ( M ); (2.5)(f) the Cheeger isoperimetric constant of M is positive [CMM1, Theorem 9.5],and consequently the following Sobolev type inequality holds: (cid:13)(cid:13) f (cid:13)(cid:13) L ≤ C (cid:13)(cid:13) |∇ f | (cid:13)(cid:13) L (2.6)for all f ∈ C ∞ c ( M ) [Ch, Theorem V.2.1].Next, we introduce the local Hardy space h ( M ). Definition 2.1.
Suppose that p is in (1 , ∞ ] and let p ′ be the index conjugate to p .A standard p -atom is a function a in L ( M ) supported in a ball B in B satisfyingthe following conditions:(i) size condition : k a k L p ≤ µ ( B ) − /p ′ ;(ii) cancellation condition : Z B a d µ = 0. FAMILY OF HARDY TYPE SPACES 5 A global p -atom is a function a in L ( M ) supported in a ball B of radius exactlyequal to p -atoms will be referred to simply as p - atoms . Definition 2.2.
The local atomic Hardy space h ,p ( M ) is the space of all func-tions f in L ( M ) that admit a decomposition of the form f = ∞ X j =1 λ j a j , (2.7)where the a j ’s are p -atoms and P ∞ j =1 | λ j | < ∞ . The norm k f k h ,p of f is theinfimum of P ∞ j =1 | λ j | over all decompositions (2.7) of f .This space was introduced in even greater generality by Volpi [Vo], who extendedprevious work of Goldberg [Go] and Taylor [T], and then further generalised in[MVo]. Goldberg treated the Euclidean case, while Taylor worked on Riemannianmanifolds with strongly bounded geometry and considered only ∞ -atoms (moreprecisely, ions); Volpi worked in a much more abstract setting, which covers thecase where M is a Riemannian manifold with Ricci curvature bounded from below(see also [MVo] for more on this). In particular, Volpi proved that h ,p ( M ) isindependent of p ; henceforth, the space h , ( M ) will be denoted simply by h ( M ),and 2-atoms in h ( M ) will also be called h ( M )-atoms.The choice of 1 as a “scale parameter” for the radii of balls in Definition 2.1 iscompletely arbitrary, and replacing it with any other positive number would leadto the definition of the the same space h ( M ), with equivalent norms. Indeed, aslight modification of [MVo, Lemma 2] shows that, for all p ∈ (1 , ∞ ], there exists aconstant C such that, for every function f in L p ( M ) supported in a ball B ∈ B \ B ,the function f is in h ( M ) and k f k h ≤ Cµ ( B ) /p ′ k f k L p . (2.8)An important feature of h ( M ) lies in its interpolation properties with L p spaces.In particular, for every θ in (0 , (cid:0) h ( M ) , L ( M ) (cid:1) [ θ ] is L / (2 − θ ) ( M ) (see [MVo, Theorem 5]).We shall repeatedly use the following proposition, whose proof is a slight modi-fication of [MVo, Theorem 6]. Proposition 2.3. If T is an h ( M ) -valued linear operator defined on h ( M ) -atomssuch that sup {k T a k h : a h ( M ) -atom } < ∞ , then T admits a unique bounded extension from h ( M ) to h ( M ) . The definition of the space h ( M ) is similar to that of the atomic Hardy space H ( M ), introduced by Carbonaro, Mauceri and Meda [CMM1, CMM2], the onlydifference being that atoms in H ( M ) are just standard atoms in h ( M ), and thereare no global atoms. As a consequence, functions in H ( M ) have vanishing integral,a property not enjoyed by all the functions in h ( M ). Thus, trivially, H ( M ) isproperly and continuously contained in h ( M ).We now introduce the space bmo ( M ). Suppose that q is in [1 , ∞ ). For eachlocally integrable function g define the local sharp maximal function g ♯,q by g ♯,q ( x ) = sup B ∈ B ( x ) (cid:16) µ ( B ) Z B | g − g B | q d µ (cid:17) /q ∀ x ∈ M, where g B denotes the average of f over B and B ( x ) denotes the family of all ballsin B centred at the point x . Define the modified local sharp maximal function A. MARTINI, S. MEDA, AND M. VALLARINO N q ( g ) by N q ( g )( x ) := g ♯,q ( x ) + h µ ( B ( x )) Z B ( x ) | g | q d µ i /q ∀ x ∈ M .
Denote by bmo q ( M ) the space of all locally integrable functions g such that N q ( g )is in L ∞ ( M ), endowed with the norm k g k bmo q = k N q ( g ) k L ∞ . In [MVo] it is proved that the space bmo q ( M ) does not depend on the parameter q , as long as q is in [1 , ∞ ). Henceforth, we shall denote this space by bmo ( M ),endowed with the norm bmo . Moreover, the space bmo ( M ) may be identifiedwith the dual of h ( M ) (see [MVo, Theorem 2]). More precisely, for every function g ∈ bmo ( M ), the linear functional F g , defined on every h ( M )-atom a by F g ( a ) = Z M a g d µ, (2.9)extends to a bounded linear functional on h ( M ). Conversely, for every functional F ∈ ( h ( M )) ′ there exists a function g ∈ bmo ( M ) such that F = F g . Moreover,there exists a positive constant C such that C − k g k bmo ≤ k F k ( h ) ′ ≤ C k g k bmo . (2.10)3. The heat semigroup and the operator U on h ( M )The theory of Hardy type spaces that we shall describe in Section 4 requires theboundedness on h ( M ) of various functions of the Laplace–Beltrami operator L ,including the heat semigroup and the operator U defined below. These will beestablished in Subsections 3.1 and 3.2, respectively.3.1. The heat semigroup on h ( M ) . It is well known that (cid:8) H t (cid:9) is a Markoviansemigroup. In particular, it is contractive on L ( M ), hence from h ( M ) to L ( M ),for h ( M ) is continuously imbedded in L ( M ) with norm ≤
1. In this sectionwe discuss the boundedness of the heat semigroup { H t } on the local Hardy space h ( M ).A well known result obtained independently by Grigor’yan and Saloff-Coste [SC,Theorem 5.5.1] says that the conjunction of the local doubling condition and thelocal Poincar´e inequality is equivalent to a local Harnack inequality for positivesolutions to the heat equation. In particular, for all R > C such that for all balls B = B ( c B , r B ), with r B < R , and for any smooth positivesolution u of ( ∂ t + L ) u = 0 in the cylinder Q := (cid:0) s − r B , s (cid:1) × B , the followinginequality holds: sup Q − u ≤ C inf Q + u, (3.1)where Q − := (cid:16) s − r B , s − r B (cid:1) × B ( c B , r B /
2) and Q + := (cid:16) s − r B , s (cid:17) × B ( c B , r B / ω in [0 , π ], we denote by S ω the half line (0 , ∞ ) if ω = 0, and the sector (cid:8) z ∈ C : z = 0 , and (cid:12)(cid:12) arg z (cid:12)(cid:12) < ω (cid:9) if ω >
0. Recall that, given a number ω in [0 , π ),an operator A on a Banach space Y is sectorial of angle ω if(i) the spectrum of A is contained in the closed sector S ω ;(ii) the following resolvent estimate holds:sup λ ∈ C \ S ω ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ ( λ − A ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y < ∞ ∀ ω ′ ∈ ( ω, π ) . FAMILY OF HARDY TYPE SPACES 7
Observe that condition (ii) above may be reformulated as follows:sup λ ∈ S ω ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ ( λ + A ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y < ∞ ∀ ω ′ ∈ [0 , π − ω ) . (3.2)The theory of Hardy type spaces that we shall develop in Section 4 hinges on theuniform boundedness of the heat semigroup on h ( M ). This fact, together withsome related estimates, will be proved in the next theorem. A result similar toTheorem 3.1 (i) below, but in a different setting, may be found in [DW]. Theorem 3.1.
The following hold: (i) { H t } is a uniformly bounded C semigroup on h ( M ) ; (ii) L is a sectorial operator of angle π/ on h ( M ) ; (iii) sup λ> (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ ( λ + L ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h < ∞ and sup λ> (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ( λ + L ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h < ∞ .Proof. First we prove (i). We shall preliminarily show thatsup t> sup (cid:12)(cid:12) h H t a, g i (cid:12)(cid:12) < ∞ , (3.3)where the inner supremum is taken over all h ( M )-atoms a and all bmo ( M )-functions g with (cid:13)(cid:13) g (cid:13)(cid:13) bmo ≤
1. In light of (2.10) and Proposition 2.3, estimate(3.3) implies the uniform boundedness of { H t } on h ( M ).To prove (3.3), let a be an h ( M )-atom supported in a ball B = B ( c B , r B ), with r B ≤
1. Denote by M a 1-discretisation of M and, for each z in M , denote by B z the ball with centre z and radius 1. It is a well known fact (see, for instance,[MVo]) that the cover { B z : z ∈ M } has the finite overlapping property. Denoteby { ψ z : z ∈ M } a partition of unity subordinate to that cover. Clearly, at leastformally, (cid:12)(cid:12) h H t a, g i (cid:12)(cid:12) ≤ X z ∈ M Z B z ψ z ( x ) (cid:12)(cid:12) H t a ( x ) (cid:12)(cid:12) (cid:12)(cid:12) g ( x ) (cid:12)(cid:12) d µ ( x ) ≤ X z ∈ M (cid:13)(cid:13) H t a (cid:13)(cid:13) L ( B z ) (cid:13)(cid:13) g (cid:13)(cid:13) L ( B z ) . We have used the Cauchy–Schwarz inequality in the last inequality above. Noticethat (cid:13)(cid:13) g (cid:13)(cid:13) L ( B z ) = µ ( B z ) / h µ ( B z ) Z B z | g | d µ i / ≤ µ ( B z ) / (cid:13)(cid:13) g (cid:13)(cid:13) bmo . Furthermore, if h t denotes the heat kernel, then (cid:13)(cid:13) H t a (cid:13)(cid:13) L ( B z ) = h Z B z d µ ( x ) (cid:12)(cid:12)(cid:12) Z B h t ( x, y ) a ( y ) d µ ( y ) (cid:12)(cid:12)(cid:12) i / ≤ h Z B z d µ ( x ) Z B h t ( x, y ) (cid:13)(cid:13) a (cid:13)(cid:13) L ( B ) d µ ( y ) i / ≤ h Z B z d µ ( x ) 1 µ ( B ) Z B h t ( x, y ) d µ ( y ) i / ;we have used Schwarz’s inequality in the first inequality and the size condition on a in the second. Clearly for each x in M µ ( B ) Z B h t ( x, y ) d µ ( y ) ≤ sup y ∈ B h t ( x, y ) ≤ C inf y ∈ B h t + r B ( x, y ) ≤ C h t + r B ( x, c B ) ;we have used Harnack’s inequality in the second inequality above. Thus, (cid:13)(cid:13) H t a (cid:13)(cid:13) L ( B z ) ≤ C h Z B z h t + r B ( x, c B ) d µ ( x ) i / , which, by Harnack’s inequality, is dominated by C µ ( B z ) / h t + r B +1 ( z, c B ). Bycombining the preceding estimates with the finite overlapping property of the family A. MARTINI, S. MEDA, AND M. VALLARINO of balls { B z : z ∈ M } , we see that (cid:12)(cid:12) h H t a, g i (cid:12)(cid:12) ≤ C (cid:13)(cid:13) g (cid:13)(cid:13) bmo X z ∈ M µ ( B z ) h t + r B +1 ( z, c B ) ≤ C (cid:13)(cid:13) g (cid:13)(cid:13) bmo X z ∈ M Z B z h t + r B +2 ( x, c B ) d µ ( x ) ≤ C (cid:13)(cid:13) g (cid:13)(cid:13) bmo Z M h t + r B +2 ( x, c B ) d µ ( x )= C (cid:13)(cid:13) g (cid:13)(cid:13) bmo , as required. The equality above follows from the Markovianity of the heat semi-group.In order to conclude the proof of (i), it remains to show that { H t } is stronglycontinuous on h ( M ). Notice that it suffices to prove that (cid:13)(cid:13) H t a − a (cid:13)(cid:13) h → t → + for every h ( M )-atom a . Indeed, suppose that this holds, and assume that f = X j c j a j . Then H t (cid:16) ∞ X j =1 c j a j (cid:17) − ∞ X j =1 c j a j = ∞ X j =1 c j (cid:0) H t a j − a j (cid:1) , because we already know that { H t } is bounded on h ( M ). Hence (cid:13)(cid:13) H t f − f (cid:13)(cid:13) h ≤ ∞ X j =1 (cid:12)(cid:12) c j (cid:12)(cid:12) (cid:13)(cid:13) H t a j − a j (cid:13)(cid:13) h → t → + by the Lebesgue dominated convergence theorem (we have already provedthat { H t } is uniformly bounded, so that (cid:13)(cid:13) H t a j − a j (cid:13)(cid:13) h ≤ t> (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h ).Now we show that, if a is an h ( M )-atom, then (cid:13)(cid:13) H t a − a (cid:13)(cid:13) h →
0. Withoutloss of generality, we may assume that the support of a is contained in a ball B centred at o with radius R ≤
1. Let D = √ L − b , and observe that, by the spectraltheorem, s cos( s D ) a is an L -valued continuous function on R , and that it takesvalue a at 0. At least on L ( M ), H t a = e − bt Z ∞−∞ h R t ( s ) cos( s D ) a d s, where h R t ( s ) = (4 πt ) − / e − s / (4 t ) is the one-dimensional Euclidean heat kernel.Write 1 = ω + ∞ X k =1 ω k , where supp ω ⊆ [ − , ω k ⊆ [ − k − , − k ] ∪ [ k, k + 1]. We are led to consider the integrals I kt := e − bt Z ∞−∞ h R t ( s ) ω k ( s ) cos( s D ) a d s , where k is a nonnnegative integer.Consider first the case where k >
0. Clearly (cid:13)(cid:13) cos( s D ) a (cid:13)(cid:13) L ≤ (cid:13)(cid:13) a (cid:13)(cid:13) L ≤ µ ( B ) − / , and moreover supp(cos( s D ) a ) ⊆ B k +2 ( o ) by finite propagation speed; hence, by(2.8) and (2.1), we conclude that (cid:13)(cid:13) cos( s D ) a (cid:13)(cid:13) h ≤ C µ (cid:0) B k +2 ( o ) (cid:1) / µ ( B ) / ≤ C k ν e βk ∀ s : k ≤ | s | < k + 1 , FAMILY OF HARDY TYPE SPACES 9 where ν is an appropriate nonnegative number. Remember that a is fixed in thisargument, so that we do not care about the factor µ ( B ) − / . Therefore (cid:13)(cid:13) I kt (cid:13)(cid:13) h ≤ C k ν e βk Z k +1 k h R t ( s ) d s ≤ C t − / k ν e βk − k / (4 t ) . It is straightforward to check that t − / k ν e βk − k / (4 t ) ≤ e − c/t − ck for some suitableconstant c and all t small enough, uniformly in k . Thus, (cid:13)(cid:13)(cid:13) ∞ X k =1 I kt (cid:13)(cid:13)(cid:13) h ≤ C ∞ X k =1 e − c/t − ck which tends to 0 as t → + .Thus, it remains to estimate (cid:13)(cid:13) I t − a (cid:13)(cid:13) h . Notice that t I t − a is an L -valuedcontinuous function on R . First, we show that I t − a is weakly convergent to 0 in L ( M ). Indeed, ( a, ψ ) L ( M ) = (cid:10) (cos( · D ) a, ψ ) L ( M ) , δ (cid:11) R , where h· , ·i R denotes thepairing between measures and continuous functions on R . Thus, if m denotes theLebesgue measure on R , (cid:0) I t − a, ψ (cid:1) L ( M ) = e − bt Z ∞−∞ h R t ( s ) ω ( s ) (cid:0) cos( s D ) a, ψ ) L d s − ( a, ψ ) L = (cid:10)(cid:0) cos( · D ) a, ψ (cid:1) L , e − bt h R t ω d m − δ (cid:11) R , which tends to 0 as t → + , because the measure e − bt h R t ω d m − δ is weaklyconvergent to 0. Furthermore, (cid:13)(cid:13) I t (cid:13)(cid:13) L ≤ e − bt Z ∞−∞ h R t ( s ) ω ( s ) (cid:13)(cid:13) cos( s D ) a (cid:13)(cid:13) L ( M ) d s ≤ (cid:13)(cid:13) a (cid:13)(cid:13) L , so that lim sup t → + (cid:13)(cid:13) I t (cid:13)(cid:13) L ≤ (cid:13)(cid:13) a (cid:13)(cid:13) L . By a well known result [B, Proposition 3.32], I t → a strongly in L ( M ).Note that the support of I t is contained in B ( o ). Hence, by (2.8), (cid:13)(cid:13) I t − a (cid:13)(cid:13) h ≤ C µ (cid:0) B ( o ) (cid:1) / (cid:13)(cid:13) I t − a (cid:13)(cid:13) L → t → + , as required to conclude the proof of the strong continuity of { H t } on h ( M ).As explained in [Haa, Section 2.1.1, p. 24], (ii) is an immediate consequence of(i) and the Hille–Yosida theorem.The first statement in (iii) follows directly from the sectoriality of L proved in(ii). To prove the last statement in (iii) observe that, by spectral theory, L ( λ + L ) − = I − λ ( λ + L ) − , at least on L ( M ). The required estimate follows from this and the first statement. (cid:3) Remark . It is worth pointing out that the heat semigroup is not uniformlybounded on H ( D ), where D denotes the hyperbolic disk. Indeed, arguing as in[Ce, Section 2.7], one can show that there exists a positive constant c such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H ( D ) ≥ c (1 + t ) ∀ t ∈ R + . We omit the proof, because the result above is not essential for the theory of Hardytype spaces developed in this paper and the details are somewhat long and intricate.This motivates the introduction of the spaces X γ ( M ) in Section 4 and explains whywe do not base our analysis on the spaces X k ( M ) introduced in [MMV2]. The operator U on h ( M ) . A central role in what follows will be played bythe family (cid:8) U σ := L ( σ + L ) − : σ > (cid:9) of (spectrally defined) operators.Clearly U σ is bounded on L ( M ), by the spectral theorem. A straightforwardconsequence of the fact that L generates the contraction semigroup { H t } on L p ( M )for every p in [1 , ∞ ] is that U σ extends to a bounded operator on L p ( M ) for allsuch values of p . Furthermore, for each λ > U σ is an isomorphismof L p ( M ), 1 < p ≤ b >
0, hence the bottom of the L p ( M ) spectrumof L is positive), and it is injective on L ( M ) [MMV2, Proposition 2.4]. We shalloften write U instead of U .The sectoriality of L on h ( M ) given by Theorem 3.1 implies the followingproperties of the operators U σ and their fractional powers. Proposition 3.3.
Assume that σ > . (i) U σ is an injective bounded sectorial operator of angle π/ on h ( M ) . (ii) U γσ is injective and bounded on h ( M ) for all γ ∈ C with Re γ > .Proof. First we prove (i). Since U σ is injective on L ( M ) it is also injective on h ( M ). By Theorem 3.1 (ii), L is a sectorial operator of angle π/ h ( M ). By[Haa, Proposition 2.1.1 (f)] so is U σ , and moreover the boundedness of U σ followsfrom Theorem 3.1 (iii).Property (ii) immediately follows from (i) and [Haa, Proposition 3.1.1]. (cid:3) Remark . The condition Re γ > U iuσ , for u real and σ >
0, may be unbounded from h ( M )to L ( M ) — and, a fortiori , on h ( M ). Indeed, suppose for instance that M is acomplex symmetric space of the noncompact type. We argue by contradiction. If U iuσ were bounded from h ( M ) to L ( M ) for some u = 0, then so would be theoperator L iu , because L iu = U iuσ (cid:0) σ + L (cid:1) iu and (cid:0) σ + L (cid:1) iu is bounded on h ( M )[MVo, Theorem 7]. However, it is known [MMV4] that L iu does not map H ( M )to L ( M ). Since H ( M ) is contained in h ( M ), L iu does not map h ( M ) to L ( M )either.An important consequence of sectoriality of an operator A on a Banach spaceis the boundedness of certain holomorphic functions of A . More precisely, supposethat 0 < θ ≤ π . We denote by H ∞ ( S θ ) the space of all bounded holomorphicfunctions on the sector S θ for which there exist positive constants C and s suchthat (cid:12)(cid:12) f ( z ) (cid:12)(cid:12) ≤ C | z | s | z | s . ∀ z ∈ S θ ; H ∞ ( S θ ) is called the Riesz–Dunford class on S θ . The extended Riesz–Dunford class E ( S θ ) is the Banach algebra generated by H ∞ ( S θ ), the constant functions and thefunction z (1 + z ) − . For more on these classes of functions, see [Haa, pp. 27–29]. Recall that if A is a sectorial operator of angle ω on a Banach space Y and f belongs to the extended Riesz–Dunford class E ( S θ ) for some θ > ω , then f ( A ) isbounded on Y [Haa, Theorem 2.3.3].The functional calculus for sectorial operators is used in the proof of the followingproposition, which contains additional information on the operators U σ . Proposition 3.5.
Assume that σ , σ , γ > . (i) U γσ (cid:2) h ( M ) (cid:3) = U γσ (cid:2) h ( M ) (cid:3) . (ii) There exists a constant C such that C − (cid:13)(cid:13) U − γσ f (cid:13)(cid:13) h ≤ (cid:13)(cid:13) U − γσ f (cid:13)(cid:13) h ≤ C (cid:13)(cid:13) U − γσ f (cid:13)(cid:13) h for every f in U γ (cid:2) h ( M ) (cid:3) . FAMILY OF HARDY TYPE SPACES 11
Proof.
It is easily checked that the function ϕ defined by ϕ ( z ) = (cid:18) z + σ z + σ (cid:19) γ belongs to the class E ( S θ ) for all θ ∈ (0 , π ). Since U − γσ U γσ = ( σ + L ) γ ( σ + L ) − γ = ϕ ( L )on L ( M ), and L is sectorial of angle π/ h ( M ) by Theorem 3.1 (ii), weconclude by [Haa, Theorem 2.3.3] and Proposition 2.3 that U − γσ U γσ extends toa bounded operator on h ( M ). Similarly one shows that U − γσ U γσ extends to abounded operator on h ( M ). Consequently the identities (cid:0) U − γσ U γσ (cid:1) (cid:0) U − γσ U γσ (cid:1) = J = (cid:0) U − γσ U γσ (cid:1) (cid:0) U − γσ U γσ (cid:1) , (3.4) U γσ = U γσ (cid:0) U − γσ U γσ (cid:1) , U γσ = U γσ (cid:0) U − γσ U γσ (cid:1) , (3.5)initially valid on L ( M ), extend by density and boundedness to h ( M ). From(3.4) we deduce that the extensions of U − γσ U γσ and U − γσ U γσ are isomorphisms of h ( M ), and from this and (3.5) it follows that U σ (cid:2) h ( M ) (cid:3) = U σ (cid:2) h ( M ) (cid:3) . Thisproves (i).From (3.5) we also deduce that, for all f ∈ U γ (cid:2) h ( M ) (cid:3) , U − γσ f = (cid:0) U − γσ U γσ (cid:1) U − γσ f, U − γσ f = (cid:0) U − γσ U γσ (cid:1) U − γσ f, and the h ( M )-boundedness of U − γσ U γσ and U − γσ U γσ gives (ii). (cid:3) A one-parameter family of Hardy type spaces
Definition and properties of X γ ( M ) . By Proposition 3.3 (ii), the operator U γ is bounded and injective on h ( M ) for all γ >
0. Thus, the following definitionmakes sense.
Definition 4.1.
Suppose that γ >
0. We denote by X γ ( M ) the space U γ (cid:2) h ( M ) (cid:3) ,endowed with the norm that makes U γ an isometry, i.e., set (cid:13)(cid:13) f (cid:13)(cid:13) X γ := (cid:13)(cid:13) U − γ f (cid:13)(cid:13) h ∀ f ∈ U γ (cid:2) h ( M ) (cid:3) . The following proposition gives some equivalent characterisations of the spaces X γ ( M ), showing in particular that replacing U with U σ for some σ > Proposition 4.2.
Let γ, σ > . For a function f on M , the following are equiva-lent: (i) f is in X γ ( M ) ; (ii) f is in U γσ (cid:2) h ( M )] ; (iii) both f and L − γ f are in h ( M ) .Moreover there exists a positive constant C independent of f such that C − (cid:13)(cid:13) f (cid:13)(cid:13) X γ ≤ (cid:13)(cid:13) U − γσ f (cid:13)(cid:13) h ≤ C (cid:13)(cid:13) f (cid:13)(cid:13) X γ , (4.1) C − (cid:13)(cid:13) f (cid:13)(cid:13) X γ ≤ (cid:13)(cid:13) f (cid:13)(cid:13) h + (cid:13)(cid:13) L − γ f (cid:13)(cid:13) h ≤ C (cid:13)(cid:13) f (cid:13)(cid:13) X γ . (4.2) Proof.
The equivalence of (i) and (ii) and the inequalities (4.1) are immediate con-sequences of Proposition 3.5. It remains to prove the equivalence of (i) and (iii), aswell as the inequalities (4.2).Assume first that both f and L − γ f belong to h ( M ). Observe that, at leastformally, U − γ f = ϕ ( L ) (cid:0) I + L − γ (cid:1) f, (4.3) where ϕ is given by ϕ ( z ) = (1 + z ) γ z γ . Now, f and L − γ f are in h ( M ) by assumption, whence so is (cid:0) I + L − γ (cid:1) f . On theother hand, it is straightforward to check that ϕ belongs to the class E ( S θ ) for any θ ∈ ( π/ , π ). Since L is a sectorial operator of angle π/ h ( M ) by Theorem3.1 (ii), we deduce that ϕ ( L ) is bounded on h ( M ). Hence from (4.3) we concludethat U − γ f is in h ( M ) and (cid:13)(cid:13) f (cid:13)(cid:13) X γ = (cid:13)(cid:13) U − γ f (cid:13)(cid:13) h ≤ C (cid:0) (cid:13)(cid:13) f (cid:13)(cid:13) h + (cid:13)(cid:13) L − γ f (cid:13)(cid:13) h (cid:1) . Conversely, suppose that U − γ f is in h ( M ). Observe that f = U γ U − γ f .Since U − γ f is in h ( M ) by assumption, and U γ is bounded on h ( M ) by Propo-sition 3.3 (ii), f is in h ( M ), and (cid:13)(cid:13) f (cid:13)(cid:13) h ≤ C (cid:13)(cid:13) U − γ f (cid:13)(cid:13) h = (cid:13)(cid:13) f (cid:13)(cid:13) X γ . Furthermore, L − γ f = ( I + L ) − γ U − γ f , and ( I + L ) − γ is bounded on h ( M )by Theorem 3.1 (iii). Then L − γ f is in h ( M ) and (cid:13)(cid:13) L − γ f (cid:13)(cid:13) h ≤ C (cid:13)(cid:13) U − γ f (cid:13)(cid:13) h = (cid:13)(cid:13) f (cid:13)(cid:13) X γ , as required. (cid:3) The operators U γ commute with any other operator in the functional calculusof L , including resolvents and the heat semigroup. Hence, from Definition 4.1 andTheorem 3.1 one immediately obtains the following result. Corollary 4.3.
The following hold: (i) the heat semigroup is uniformly bounded on X γ ( M ) for every γ > ; (ii) L is a sectorial operator of angle π/ on X γ ( M ) ; (iii) sup λ> (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ ( λ + L ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X γ < ∞ , equivalently sup λ> (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ( λ + L ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X γ < ∞ . A few other relevant properties of the spaces X γ ( M ) are stated below. Oneshoud compare parts (iii) and (iv) of Proposition 4.4 with the discussion in Remark3.4. In light of the interpolation property in part (v), the X γ - h boundednessof the imaginary powers of L expressed in part (iv) can be seen as an endpointcounterpart to their L p -boundedness for p ∈ (1 , ∞ ). Proposition 4.4.
The following hold: (i) if Re z > , then U z is bounded on X γ ( M ) for every γ > ; (ii) { X γ ( M ) : γ > } is a decreasing family of Banach spaces; (iii) if γ > and u is real, then U iu is bounded from X γ ( M ) to h ( M ) ; (iv) if γ > and u is real, then L iu is bounded from X γ ( M ) to h ( M ) ; (v) (cid:0) X γ ( M ) , L ( M ) (cid:1) [ θ ] = L p θ ( M ) , whenever θ ∈ (0 , and p θ = 2 / (1 − θ ) .Proof. Observe that U z is bounded on X γ ( M ) if and only if (cid:2) U γ (cid:3) − U z U γ = U z is bounded on h ( M ). Thus (i) is an immediate consequence of Proposition 3.3 (ii).Next we prove (ii). By Proposition 3.3 (ii) and the definition of X γ ( M ), it is clearthat h ( M ) ⊇ X γ ( M ) for any γ >
0, with continuous inclusion. So, if γ > γ > X γ ( M ) = U γ (cid:2) h ( M ) (cid:3) = U γ − γ U γ (cid:2) h ( M ) (cid:3) = U γ − γ (cid:2) X γ ( M ) (cid:3) ⊆ X γ ( M ) , the last containment above being a consequence of the boundedness of U γ − γ on X γ ( M ) proved in (i).Notice that (iii) is equivalent to the boundedness of U γ + iu on h ( M ), so (iii) isanother consequence of 3.3 (ii). FAMILY OF HARDY TYPE SPACES 13
To prove (iv), notice that, by Proposition 4.2, L iu is bounded from X γ ( M ) to h ( M ) if and only if L iu U γσ is bounded on h ( M ) for some σ >
0. Note that, byspectral theory, L iu U γσ = U γ + iuσ (cid:0) σ + L (cid:1) iu ;the operator U γ + iuσ is bounded on h ( M ) by (iii), and (cid:0) σ + L (cid:1) iu is bounded on h ( M ) because its symbol satisfies a Mihlin–H¨ormander condition of any order onthe strip { z ∈ C : | Im z | < β } provided σ > β − b [MVo, Theorem 7], and (iv) isproved.Finally, as already mentioned, U is an isomorphism of L p ( M ) for all p ∈ (1 , U γ , while X γ ( M ) = U γ (cid:2) h ( M ) (cid:3) , for all γ >
0. The interpolationproperty (v) for X γ ( M ) is therefore an immediate consequence of the correspondingproperty for h ( M ) proved in [MVo, Theorem 5]. (cid:3) We shall show that { X γ ( M ) : γ > } is actually a strictly decreasing family ofBanach spaces. To do so, we need to discuss an atomic decomposition of the spaces X γ ( M ).4.2. Atomic decomposition when γ is an integer. In the case where γ isan integer, X γ ( M ) was defined in [Vo], where also some of its properties were in-vestigated. In particular, it was shown there that these spaces admit an atomicdecomposition that we now describe, which is a variant of the atomic decompo-sition for the spaces X k ( M ) proved in [MMV3]. An atom A in X k ( M ) will be astandard atom in h ( M ) satisfying an additional infinite dimensional cancellationcondition, expressed as orthogonality of A to the space of k -harmonic functions ina neighbourhood of the support of A . Definition 4.5.
Suppose that k is a positive integer and that B is a ball in M .We say that a function V in L ( M ) is k -harmonic on B if L k V is zero (in thesense of distributions) in a neighbourhood of B . We shall denote by P kB the space of k -harmonic functions on B . Moreover, let Q kB denote the space of k -quasi-harmonicfunctions on B , i.e., the subspace of L ( M ) consisting of all the functions V suchthat L k V is constant (in the sense of distributions) in a neighbourhood of B . Remark . By elliptic regularity, P kB coincides with the space of the functions V in L ( M ) that are smooth in a neighbourhood of B and such that L k V is zerotherein. A similar remark applies to Q kB .A direct consequence of the definition of P kB and Q kB is the following chain ofinclusions: P B ⊆ Q B ⊆ P B ⊆ Q B ⊆ · · · ;correspondingly ( P B ) ⊥ ⊇ ( Q B ) ⊥ ⊇ ( P B ) ⊥ ⊇ ( Q B ) ⊥ ⊇ · · · , where ( P kB ) ⊥ and ( Q kB ) ⊥ denote the orthogonal complements of P kB and Q kB is L ( M ).For each ball B in M , let us denote by L ( B ) the space of all L ( M ) functionssupported in B . The following result is the counterpart for the spaces P kB of [MMV3,Proposition 3.3], where the case of Q kB is treated; the proof is analogous and isomitted. Proposition 4.7.
Suppose that k is a positive integer, and that B is a ball in M . (i) ( P kB ) ⊥ = (cid:8) F ∈ L ( M ) : L − k F ∈ L ( B ) (cid:9) . (ii) L − k (cid:0) ( P kB ) ⊥ ) is contained in L ( B ) ∩ Dom( L k ) . Furthermore, functionsin ( P kB ) ⊥ have support contained in B . (iii) U − k (cid:0) ( P kB ) ⊥ ) is contained in L ( B ) . Definition 4.8.
Suppose that k is a positive integer. A standard X k -atom associ-ated to the ball B of radius ≤ A in L ( M ), supported in B , suchthat(i) A is in ( Q kB ) ⊥ ;(ii) k A k L ≤ µ ( B ) − / .A global X k -atom associated to the ball B of radius 1 is a function A in L ( M ),supported in B , such that(i) A is in ( P kB ) ⊥ ;(ii) k A k L ≤ µ ( B ) − / .An X k - atom is either a standard X k -atom or a global X k - atom . Remark . An X k -atom (standard or global) is also a standard h -atom: indeed,in either case the cancellation condition (i) implies that the integral of A vanishes,since χ B is in P kB and in Q kB . Remark . The set of X k -atoms is a bounded subset of X k ( M ). Indeed, in thecase of a global X k -atom A , from Proposition 4.7 and (2.8) it follows immediatelythat k U − k A k h ≤ C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) U − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ;the fact that the same estimate also holds for a standard X k -atom A can be shownas in [MMV3, Remark 3.5]. Definition 4.11.
Suppose that k is a positive integer. The space X k at ( M ) is thespace of all functions F in h ( M ) that admit a decomposition of the form F = P j λ j A j , where { λ j } is a sequence in ℓ and { A j } is a sequence of X k -atoms. Weendow X k at ( M ) with the norm k F k X k at = inf nX j | λ j | : F = X j λ j A j , A j X k -atoms o . From Remark 4.10 it is clear that X k at ( M ) ⊆ X k ( M ), with continuous embedding.One can show that equality holds under a suitable geometric hypothesis on M . Definition 4.12.
We say that M has C ℓ bounded geometry if the injectivity radiusis positive and the following hold:(a) if ℓ = 0, then the Ricci tensor is bounded from below;(b) if ℓ is positive, then the covariant derivatives ∇ j Ric of the Ricci tensor areuniformly bounded on M for all j ∈ { , . . . , ℓ } .The aforementioned atomic decomposition of X k ( M ) is the content of the fol-lowing theorem. We omit the proof, which follows the lines of the proof of [MMV3,Theorem 4.3]. Theorem 4.13.
Suppose that k is a positive integer and that M has C k − boundedgeometry. Then X k ( M ) and X k at ( M ) agree as vector spaces and there exists a con-stant C such that C − k F k X k at ≤ k F k X k ≤ C k F k X k at ∀ F ∈ X k ( M ) . (4.4)Notice that when k = 1 the geometric hypothesis of Theorem 4.13 is alreadycontained in our geometric assumptions on the manifold M (see Section 2), so theatomic characterization of X ( M ) holds without additional assumptions.As a consequence of the atomic characterization of the space X ( M ) we can provea result involving all X γ ( M ) spaces. Denote by H ∞ ( M ) the space of all bounded FAMILY OF HARDY TYPE SPACES 15 harmonic functions on M , thought of as a subspace of bmo ( M ), and consider theannihilator of H ∞ ( M ) in h ( M ), defined by H ∞ ( M ) ⊥ := (cid:8) f ∈ h ( M ) : h f, H i = 0 for all H in H ∞ ( M ) (cid:9) , where h· , ·i denotes the duality between h ( M ) and bmo ( M ) (see (2.9)). Proposition 4.14.
The following hold: (i) for every γ > the space X γ ( M ) is contained in H ∞ ( M ) ⊥ ; (ii) { X γ ( M ) : γ > } is a strictly decreasing family of Banach spaces.Proof. We first show that X ( M ) ⊆ H ∞ ( M ) ⊥ . By Theorem 4.13, every function f in X ( M ) admits a representation of the form P j c j A j , where the A j ’s are X -atoms. Each of these is annihilated by all bounded harmonic functions, so that, by(2.9), h f, H i = X j c j Z M A j H d µ = 0 ∀ H ∈ H ∞ ( M ) . Taking closures in h ( M ), we then obtain that X ( M ) h ( M ) ⊆ H ∞ ( M ) ⊥ . Now,the sectoriality of U implies [Haa, Proposition 3.1.1 (d)] that X ( M ) h ( M ) = X γ ( M ) h ( M ) , and we can conclude that X γ ( M ) h ( M ) ⊆ H ∞ ( M ) ⊥ . This proves(i).We now prove (ii). Let γ > γ >
0. In view of Proposition 4.4 it remainsto show that the containment X γ ( M ) ⊆ X γ ( M ) is proper. We argue by con-tradiction. Suppose that X γ ( M ) = X γ ( M ). Since U is an injective sectorialoperator on h ( M ) (see Proposition 3.3), (cid:0) U γ (cid:1) − = (cid:0) U − (cid:1) γ = U − γ [Haa,Propositions 3.1.1 (e) and 3.2.1 (a)]. Furthermore, U − γ U γ ⊆ U γ − γ [Haa,Proposition 3.2.1 (b)]. Since X γ ( M ) = X γ ( M ), the operator U − γ U γ is surjec-tive on h ( M ), so U γ − γ is also surjective on h ( M ). By [Haa, Proposition 3.1.1(d)], U γ − γ is bounded and injective on h ( M ), whence U γ − γ is an isomor-phism of h ( M ). Therefore 0 is in the resolvent set of U γ − γ . However, 0 is in the h -spectrum of U ( U cannot be surjective, because, by (i), U h ( M ) = X ( M ) iscontained in the annihilator of constant functions), and this contradicts the spectralmapping theorem [Haa, Proposition 3.1.1 (j)]. (cid:3) Lack of atomic decomposition when γ is not an integer. In this sub-section we restrict our analysis to symmetric spaces of the noncompact type X ofreal rank one, and show that, if γ is not a positive integer, then X γ ( X ) does notadmit an atomic decomposition. A similar result holds for the analogue of X γ ( X )on homogeneous trees (see [CM, Theorem 5.8] for details). For the notation and themain properties of noncompact symmetric spaces and for spherical analysis thereonwe refer the reader to [H1, H2].We recall here that X is a quotient G/K , where G is a noncompact semisimpleLie group of finite centre and real rank one and K is a maximal compact subgroupof G . Given a Cartan decomposition g = p ⊕ k of the Lie algebra of G , we denote by a a maximal abelian subspace of p and by a ∗ C the complexification of its dual. Thereal rank one assumption means that dim a = 1 and implies that the Weyl group is { , − } . If M denotes the centralizer of a in K , we denote by B the quotient K/M .For every compactly supported function f on X , its Helgason–Fourier transform e f is a function on a ∗ C × B defined as in [H2, p. 223]. The Paley–Wiener theorem onnoncompact symmetric spaces will be a key ingredient in the proof of the followingresult. Theorem 4.15.
Suppose that X is a symmetric space of the noncompact type andreal rank one, that k is a positive integer, and that γ is in ( k − , k ) . (i) If f is a function in X γ ( X ) with compact support, then f belongs to X k ( X ) . (ii) If f is a function in L ( X ) ∩ X γ ( X ) with compact support contained in theball B ( o ) , then f is a multiple of an X k -atom. (iii) X k ( X ) is not dense in X γ ( X ) .Proof. We prove the result in the case where k = 1. The proof for k ≥ γ is in (0 , f is a function in X γ ( X )with compact support, then f belongs to X ( X ). Suppose that the support of f iscontained in the ball B R ( o ) for some R >
0. Then, by the Paley–Wiener theoremfor the Helgason–Fourier transform [H2, Corollary 5.9, p. 281] and the fact that f is in L ( X ), e f ( · , b ) extends to an entire function of exponential type R uniformly in b and there exist constants C and N such that (cid:12)(cid:12) e f ( λ, b ) (cid:12)(cid:12) ≤ C (1 + | λ | ) N e | Im λ | R ∀ λ ∈ a ∗ C ∀ b ∈ B. (4.5)Furthermore, e f is smooth on a ∗ C × B and satisfies the following symmetry condition: Z B e ( − iλ + ρ )( A ( x,b )) e f ( − λ, b ) d b = Z B e ( iλ + ρ )( A ( x,b )) e f ( λ, b ) d b (4.6)for every x in X and every λ in a ∗ C ; here ρ ∈ a ∗ denotes as usual half the sum of thepositive restricted roots, while A : X × B → a is defined as in [H2, p. 223].Since f is in X γ ( M ), by Proposition 4.2 there exists a function g in h ( X ) suchthat f = L γ g. Since g is in L ( X ), its Helgason–Fourier transform e g ( · , b ) is a continuous functionon a ∗ + i [ − , ρ for almost all b in B [H3, SS], and e f ( λ, b ) = Q ( λ ) γ e g ( λ, b ) , (4.7)where Q is the quadratic form on a ∗ C × a ∗ C defined by Q ( λ ) = h λ, λ i + h ρ, ρ i (see [H2, Lemma 1.4, p. 225] and [CGM, Section 1]). Note that Q vanishes at thepoints of ± iρ ; hence from (4.7) we deduce that e f ( ± iρ, b ) = 0 for almost all b in B ,and actually this holds for all b in B because e f is smooth on a ∗ C × B . Since e f ( · , b )is entire, its zeros must have at least order 1. Moreover 1 /Q is a meromorphicfunction in a ∗ C with simple poles at ± iρ . Therefore λ Q ( λ ) − e f ( λ, b ) is an entirefunction for every b in B . Since Q ( − λ ) = Q ( λ ) for all λ in a ∗ C , from the symmetrycondition (4.6) we deduce that Z B e ( − iλ + ρ ) A ( x,b ) Q ( − λ ) − e f ( − λ, b ) d b = Z B e ( iλ + ρ ) A ( x,b ) Q ( λ ) − e f ( λ, b ) d b. Furthermore, ( λ, b ) Q ( λ ) − e f ( λ, b ) is clearly smooth on a ∗ C × B , and satisfies theestimate (4.5) (possibly with a different constant C ). Again by the Paley–Wienertheorem for the Helgason–Fourier transform, there exists a distribution h on X supported in B R ( o ) such that e h ( λ, b ) = Q ( λ ) − e f ( λ, b ) , that is, h = L − f . By assumption f ∈ X γ ( X ) ⊆ h ( X ) ⊆ L ( X ). Hence, by[CGM, Theorem 4.7], L − f is a function in L p ( X ) for every p ∈ (cid:0) , n/ ( n − (cid:1) .Since L − f is supported in B R ( o ), by (2.8) we deduce that L − f ∈ h ( X ). ByProposition 4.2 we conclude that f belongs to X ( X ).We now prove (ii). Suppose that f ∈ L ( X ) ∩ X γ ( X ) and has support containedin the ball B = B ( o ). Then f is integrable, so from the proof of (i) it follows that FAMILY OF HARDY TYPE SPACES 17 L − f is supported in B , and moreover L − f is in L ( M ) because L − is L -bounded. From Proposition 4.7 (i) we deduce that f is in ( P B ) ⊥ , and consequently f is a multiple of a global X -atom.Finally we prove (iii), i.e., we show that X ( X ) is not dense in X γ ( X ). Consider X ( X ) X γ ( X ) , i.e., the closure of X ( X ) in X γ ( X ). Since U − γ is an isometry between X γ ( X ) and h ( X ), the topology of X γ ( X ) is transported by U − γ to that of h ( X ),and U − γ (cid:0) X ( X ) X γ ( X ) (cid:1) = U − γ X ( X ) h ( X ) . Clearly U − γ X ( X ) = X − γ ( X ) and weobtain that U − γ (cid:0) X ( X ) X γ ( X ) (cid:1) = X − γ ( X ) h ( X ) . However, Proposition 4.4 (vi) ensures that X − γ ( X ) is properly contained in H ∞ ( X ) ⊥ ,so that X − γ ( X ) h ( X ) ⊆ H ∞ ( X ) ⊥ , which we already know to be a proper closedsubspace of h ( X ). Thus, X ( X ) X γ ( X ) ⊆ U γ H ∞ ( X ) ⊥ ( U γ h ( X ) = X γ ( X ). (cid:3) Modified Riesz–Hardy space
In this section, we consider the modified Riesz–Hardy space e H R ( M ), defined by e H R ( M ) := (cid:8) f ∈ h ( M ) : | R f | ∈ L ( M ) (cid:9) , where R denotes the “geometric” Riesz transform ∇ L − / . Clearly e H R ( M ) ⊆ H R ( M ), where H R ( M ) is the Riesz–Hardy space defined in (1.3). The main resultof this section is the characterisation of e H R ( M ) as the space X / ( M ) introducedin the previous section.An important ingredient in the proof of this characterisation of e H R ( M ) is thefollowing boundedness result. Proposition 5.1.
The operator ( σ + L ) − / is bounded from L ( M ) to h ( M ) forall σ > β − b . In the proof of this proposition we shall apply the following lemma, which isa slight variant of [MMV1, Lemma 2.4] and [MMV2, Lemma 5.1]. Let J ν ( t ) = t − ν J ν ( t ) , where J ν denotes the standard Bessel function of the first kind and order ν , and let O denote the differential operator t∂ t on the real line. Lemma 5.2.
For every positive integer N there exists a polynomial P N of degree N without constant term such that Z + ∞−∞ f ( t ) cos( tv ) d t = Z + ∞−∞ P N ( O ) f ( t ) J N − / ( tv ) d t , for all compactly supported functions f such that O ℓ f ∈ L ( R ) for every ℓ =0 , . . . , N .Proof of Proposition 5.1. Suppose that g is in L ( M ). Observe that we can write g = P j g j , where each g j is supported in a ball of radius 1 and k g k L = P j k g j k L .It suffices to prove that ( σ + L ) − / g j is in h ( M ), and that there exists a constant C , independent of g and j , such that (cid:13)(cid:13) ( σ + L ) − / g j (cid:13)(cid:13) h ≤ C (cid:13)(cid:13) g j (cid:13)(cid:13) L . For the sake of convenience, in the rest of this proof we suppress the index j and write h instead of g j . We assume that the support of h is contained in the ball B ( o ). We argue as in the proof of [MMV3, Lemma 4.2]. Denote by ω an even functionin C ∞ c ( R ) which is supported in [ − / , / − / , / X j ∈ Z ω ( t − j ) = 1 ∀ t ∈ R . Define ω := ω , and, for each j in { , , , . . . } , ω j ( t ) := ω ( t − j ) + ω ( t + j ) ∀ t ∈ R . (5.1)Observe that the support of ω j is contained in the set of all t in R such that j − / ≤ | t | ≤ j + 3 /
4. Set m ( λ ) = ( c + λ ) − / , where c := √ σ + b > β . Recallthat b m ( t ) = K ( c | t | ) ∀ t ∈ R , (5.2)where K ν denotes the modified Bessel function of the third kind and order ν (see,for instance, [Le, p. 108]).As in the proof of Theorem 3.1, let D = √ L − b . By the spectral theorem, wewrite ( σ + L ) − / = m ( D ) = P ∞ j =0 T j ( D ) , where T j ( λ ) = Z ∞−∞ ( ω j b m )( t ) cos( tλ ) d t ∀ λ ∈ R . (5.3)An induction argument, based on formulae [Le, (5.7.9), p. 110] and [Le, (5.7.12),p. 111], and the estimates [Le, formulae (5.7.12) and (5.11.9)] show that, for all ℓ ∈ N , the function O ℓ K is in L ( R + ), and, for all ℓ ≥ O ℓ K is bounded;moreover, for all ε ∈ (0 ,
1) and ℓ ∈ N , there exists a positive constant C such that | O ℓ K ( t ) | ≤ C e − εt ∀ t ≥ / . (5.4)In view of (5.2), it is straightforward to check that O ℓ ( ω j b m ) is in L ( R ) for allnonnegative integers ℓ and j , and that O ℓ ( ω j b m ) is bounded whenever ℓ + j ≥ N , for all j ≥ λ ∈ R T j ( λ ) = Z + ∞−∞ P N ( O )( ω j b m )( t ) J N − / ( tλ ) d t , (5.5)where P N is a polynomial of degree N without constant term.By (5.4), for all N there exist positive constants C and c ′ ∈ ( β, c ) such that, for j = 1 , , , . . . , (cid:12)(cid:12) P N ( O )( ω j b m )( t ) (cid:12)(cid:12) ≤ C e − c ′ | t | (5.6)on the support of ω j . By the asymptotics of J N − / [Le, formula (5.11.6)],sup s> | (1 + s ) N J N − / ( s ) | < ∞ . Let k J N − / ( t D ) denote the Schwartz kernel of the operator J N − / ( t D ). If wechoose N > ( n + 2) /
2, we may apply [MMV1, Proposition 2.2 (i)] and concludethat (cid:13)(cid:13) J N − / ( t D ) h (cid:13)(cid:13) L ≤ (cid:13)(cid:13) h (cid:13)(cid:13) L sup y ∈ M (cid:13)(cid:13) k J N − / ( t D ) ( · , y ) (cid:13)(cid:13) L ≤ C (cid:13)(cid:13) h (cid:13)(cid:13) L | t | − n/ (cid:0) | t | (cid:1) n/ (5.7)for every t ∈ R \ { } . Take j ≥ P N ( O )( ω j b m ) iscontained in { t ∈ R : j − / ≤ | t | ≤ j + 3 / } . Hence (cid:13)(cid:13) T j ( D ) h (cid:13)(cid:13) L ≤ Z ∞−∞ (cid:12)(cid:12) P N ( O )( ω j b m )( t ) (cid:12)(cid:12) (cid:13)(cid:13) J N − / ( t D ) h (cid:13)(cid:13) L d t ≤ C (cid:13)(cid:13) h (cid:13)(cid:13) L Z j +3 / j − / (cid:12)(cid:12) P N ( O )( ω j b m )( t ) (cid:12)(cid:12) (cid:12)(cid:12) t (cid:12)(cid:12) − n/ (cid:0) | t | (cid:1) n/ d t (5.8) ≤ C e − c ′ j (cid:13)(cid:13) h (cid:13)(cid:13) L ∀ j ∈ { , , . . . } . FAMILY OF HARDY TYPE SPACES 19
In the last inequality we have used (5.7) and (5.6). By (5.5) and finite propagationspeed, T j ( D ) h is a function in L ( M ) with support contained in B j +1 ( o ); so, by(2.8), (cid:13)(cid:13) T j ( D ) h (cid:13)(cid:13) h ≤ C (cid:13)(cid:13) h (cid:13)(cid:13) L e − c ′ j j α/ e βj , for every j ∈ { , , . . . } . Hence (cid:13)(cid:13)(cid:13) ∞ X j =1 T j ( D ) h (cid:13)(cid:13)(cid:13) h ≤ C (cid:13)(cid:13) h (cid:13)(cid:13) L ∞ X j =1 j α/ e ( β − c ′ ) j ≤ C (cid:13)(cid:13) h (cid:13)(cid:13) L , because c ′ > β .It remains to estimate T ( D ). Observe that, for all t ∈ R \ { } , since J N − / ( t · )is entire of exponential type | t | , by finite propagation speed k J N − / ( t D ) ( · , y ) is sup-ported in B | t | ( y ). Hence, by H¨older’s inequality and [MMV1, Proposition 2.2 (i)],for every t ∈ [ − , \ { } and p ∈ [1 , (cid:13)(cid:13) J N − / ( t D ) h (cid:13)(cid:13) L p ≤ (cid:13)(cid:13) h (cid:13)(cid:13) L sup y ∈ M (cid:13)(cid:13) k J N − / ( t D ) ( · , y ) (cid:13)(cid:13) L p ≤ C (cid:13)(cid:13) h (cid:13)(cid:13) L | t | n (1 /p − / sup y ∈ M (cid:13)(cid:13) k J N − / ( t D ) ( · , y ) (cid:13)(cid:13) L ≤ C (cid:13)(cid:13) h (cid:13)(cid:13) L | t | − n/p ′ . Therefore, we see that, if p ′ > n , then (cid:13)(cid:13) T ( D ) h (cid:13)(cid:13) L p ≤ Z ∞−∞ (cid:12)(cid:12) P N ( O )( ω b m )( t ) (cid:12)(cid:12) (cid:13)(cid:13) J N − / ( t D ) h (cid:13)(cid:13) L p d t ≤ C (cid:13)(cid:13) h (cid:13)(cid:13) L Z − (cid:12)(cid:12) P N ( O )( ω b m )( t ) (cid:12)(cid:12) (cid:12)(cid:12) t (cid:12)(cid:12) − n/p ′ d t ≤ C (cid:13)(cid:13) h (cid:13)(cid:13) L ;observe that P N ( O )( ω b m ) is bounded since P N has no constant term. Again byfinite propagation speed, T ( D ) h is supported in B ( o ), so from (2.8) we deducethat (cid:13)(cid:13) T ( D ) h (cid:13)(cid:13) h ≤ C (cid:13)(cid:13) h (cid:13)(cid:13) L , as required to conclude the proof. (cid:3) Theorem 5.3.
The modified Riesz–Hardy space e H R ( M ) agrees with X / ( M ) .Proof. First we prove that X / ( M ) ⊆ e H R ( M ). Suppose that f is in X / ( M ), andlet σ >
0. Then, by Proposition 4.2, there exists g in h ( M ) such that U / σ g = f ,and therefore (cid:12)(cid:12) R f (cid:12)(cid:12) = (cid:12)(cid:12) ∇ ( σ + L ) − / g (cid:12)(cid:12) . It is well known that the local Riesz transform ∇ ( σ + L ) − / maps h ( M ) to L ( M ), provided σ is large enough [MVo, Theorem 8], hence (cid:12)(cid:12) R f (cid:12)(cid:12) belongs to L ( M ). Furthermore, f itself belongs to h ( M ), because X / ( M ) ⊆ h ( M ). Thus, f is in e H R ( M ), as required.Next we prove that e H R ( M ) ⊆ X / ( M ). Suppose that f is in e H R ( M ). Then f is in h ( M ) and (cid:12)(cid:12) R f (cid:12)(cid:12) is in L ( M ). By the inequality (2.6), L − / f is in L ( M ).Choose σ > β − b . Notice that √ σ + z √ z = ϕ ( z )+ √ σ √ z , where the function ϕ belongsto the class E ( S θ ) for all θ in (0 , π ). Since L generates a contraction semigroupon L ( M ), L is a sectorial operator of angle π/ L ( M ), and therefore ϕ ( L ) isbounded on L ( M ) [Haa, Theorem 2.3.3]. We already know that L − / f belongsto L ( M ), whence U − / σ f = ϕ ( L ) f + √ σ L − / f is in L ( M ). Now, set g = U − / σ f . Then L − / f = ( σ + L ) − / g , which belongs to h ( M )by Proposition 5.1. Since f belongs to h ( M ) by assumption, we conclude that f ∈ h ( M ) by Proposition 4.2. (cid:3) References [A] J.-Ph. Anker, Sharp estimates for some functions of the Laplacian on noncompactsymmetric spaces,
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School of Mathematics, University of Birmingham, Edgbaston, Birm-ingham, B15 2TT, United Kingdom
E-mail address : [email protected] (Stefano Meda) Dipartimento di Matematica e Applicazioni, Universit`a di Milano-Bicocca, via R. Cozzi 53, I-20125 Milano, Italy
E-mail address : [email protected] (Maria Vallarino) Dipartimento di Scienze Matematiche “Giuseppe Luigi Lagrange”,Dipartimento di Eccellenza 2018-2022, Politecnico di Torino, corso Duca degli Abruzzi24, 10129 Torino, Italy
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