Besov spaces and the boundedness of weighted Bergman projections over symmetric tube domains

Publ. Mat. 49 (2005), 21–72 BESOV SPACES AND THE BOUNDEDNESS OF WEIGHTED BERGMAN PROJECTIONS OVER SYMMETRIC TUBE DOMAINS Daniele Debertol Abstract We extend the analysis of weighted Bergman spaces Ap,q son symmetric tube domains, contained in [2], to the case where the weights are positive powers ∆s . = ∆s1−s2 1·... ·∆sr−1−sr r−1∆sr r of the principal minors ∆1,...,∆ron the symmetric cone Ω. We discuss the realization of the boundary distributions of functions in Ap,q sin terms of Besov-type spaces Bp,q sadapted to the structure of the cone. We give a necessary and a sufficient condition on the values of p,qand sfor which this identification between Ap,q s and Bp,q sholds. We also present a continuous version of these latter spaces which is new even for the case s1=··· =srconsidered in [2]. We use these results to discuss multipliers between Besov spaces and the boundedness of the weighted Bergman projection Ps:Lp,q s→ Ap,q s. The situation in the rank two case is specifically dealt with. 1. Introduction Let Ω be an irreducible symmetric cone inside a real vector space Vof dimension nendowed with the structure of a euclidean Jordan algebra with identity e. In particular, Ω is self-dual w.r.t. the inner product (x|y). = tr(xy) on V. As in [7], we shall define r. = rank(V) and ∆jto be the j-th principal minor on V,j= 1,...,r. Then, we can write Ω = {x∈V: ∆j(x)>0, j = 1,...,r}. When s= (s1,...,sr)∈Rr, the generalized power function ∆sis defined on Ω by (1.1) ∆s . = ∆s1−s2 1·...·∆sr−1−sr r−1∆sr r. 2000 Mathematics Subject Classification. 42B35, 32A25. Key words. Bergman projection, Jordan algebra, Besov multipliers, boundary values. 22 D. Debertol We refer to [7], [2] and Section 2.1 below for a more detailed description of these notions. The family of Bergman spaces Ap,q son the tube domain TΩ. =V+iΩ over the cone Ω will be defined with respect to the weights ∆s. The main concern of this paper is the boundedness of the Bergman projection Ps:Lp,q s−→ Ap,q s. This operator has an explicit kernel given by the formula Ks(z, w) = d(s)∆−s−(n r,..., n r)z−w i for some constant d(s), once ∆sfunctions have been suitably extended to TΩ(see (5.2)). The goal of this paper is twofold: first, to develop a continuous analogue of the techniques associated with Besov spaces for the cone, presented in [2], and second, to generalize some of the topics covered there in the special case s= (s1,...,sr) = (ν,...,ν) to the wider family of weighted Bergman projections Ps, when s6= (ν,...,ν). In the end, both of these aspects will be seen to rely onto the following remark: the triangular subgroup Tin the Iwasawa decomposition G=TK of the structure group Gof Ω is sufficient to perform the analysis of many of the results in [2]. That is, in this paper we show that the Kpart of G can be made to play no role whatsoever. In this context, the class of ∆sweights is the natural one to be considered: in fact, one can see that generalized powers exhaust the set of continuous and positive homomorphisms of T, see [10, 2.4], while group homomorphisms of Gmust be of the form g7→ ∆ν r(g·e). Moreover, ∆sfunctions naturally arise in related analytic issues connected to representation theory, see e.g. [7]. We begin with the study of the Bergman spaces Ap,q s(TΩ) associated to the tube domain over Ω in the complexification of V,TΩ⊂VC. These are defined to consist of the holomorphic functions Fon TΩwhich satisfy the weighted and mixed-norm integrability condition (1.2) kFkLp,q s . = ZΩZV |F(x+iy)|pdxq p ∆s(y)dy ∆(y)n r!1 q <+∞. We shall show in Theorem 2.15 that Ap,q sis non-trivial if and only if sj>(j−1) n r−1 r−1for every j= 1,...,r. Besov Spaces and Bergman Projections 23 The key tool in our analysis of Bergman projections is a continuous version of the Whitney decomposition used in [3], [1], [2], adapted to the geometric-invariant structure of the cone Ω. To introduce our framework, let T∗denote the adjoint group of T and dτ the left Haar measure on T∗; further, for b ψ∈C∞ c(Ω) and τ∈T∗, define ψτ. =F−1b ψ◦τ−1. Then, under a normalizing condition on ψ, we will show in Proposition 3.2 that for Schwartz functions fwith b fsupported in Ω the following continuous version of the Littlewood-Paley decomposition holds true: (1.3) f=ZT∗ f∗ψτdτ. One is led by (1.3) to consider the following norm on SΩ . ={f∈ S(V)| b fis supported in Ω}: (1.4) kfkBp,q s . =ZT∗ ∆s((τe)−1)kf∗ψτkq Lp(V,dx)dτ1 q , and to introduce a family of homogeneous Besov-type spaces Bp,q sas the completion of SΩw.r.t. the norm in (1.4). Note that the norm in (1.4) is defined in such a way that it enjoys the same invariance properties of the Ap,q snorm under the action of elements of T, see (2.4) and (3.17); on the other hand, the normalization chosen for the sindices is convenient in order to deal with Ap,q sspaces, but it does not always match the standard notation in the literature, for instance when n= 1, see e.g. [15]. We want to stress that it is possible to define a discrete version of the Besov spaces Bp,q sas well, following step by step the construction in [2] for the case s= (ν,...,ν). In fact, we shall show that the two versions of Besov spaces coincide, up to equivalent norms. The path we follow is to concentrate on the new presentation of Bp,q s: as a consequence, when the need occurs for results which are clear generalizations of the corresponding statements in [2], we will often quote them without proof, just adding some extra details if appropriate. The main advantage in the new presentation of Bp,q slies in the fact that it is best suited to exploit the simply transitive action of T∗on Ω, for T∗is a group, while generally the Whitney lattice underlying the discrete decomposition given in [2] is not. This will become apparent in the multiplier Theorem 3.17, for instance, and above all in the final part 24 D. Debertol of the paper, when limiting arguments based on Corollary 4.7 come into play. For the reader’s convenience, we summarize below some results about Besov spaces scattered through the paper. Most of them are straightforward generalizations of the corresponding results in [2]. Theorem 1.1. Let s∈Rr,1≤p, q ≤+∞. Then, 1) Bp,q sis a Banach space, independent of the choice of ψ, up to equivalent norms. 2) Bp,q scan be identified with the space of equivalence classes of tempered distributions on Vwith finite seminorm (1.4) and whose Fourier transform is supported in Ω, modulo S0 ∂Ω. 3) If 1< p, q < +∞, the dual space of Bp,q scan be identified with Bp0,q0 −(q0−1)sby means of the usual pairing. The continuous version of Besov spaces will also allow for a treatment of multipliers between Besov spaces which, apart being highly suited for the functional calculus of box operators s, provides a new, unified formulation of this part of the theory. The remaining part of the paper deals with the boundedness of Bergman projections. That is, we consider the orthogonal projector Ps:L2,2 s−→ A2,2 s, and we ask for the existence of bounded extensions into Lp,q sspaces. In the general case of tube domains over symmetric cones, the hint is given by the Paley-Wiener Theorem 2.7 for A2,2 s: a function F∈ A2,2 s can be written as the Fourier-Laplace transform F=Lbgof a unique distribution (actually, a locally square-integrable function) g∈B2,2 s. Note that in particular g∈ S0(V) is defined on the Bergman-Shilov boundary V× {0}of TΩ. Therefore, we plan to exploit the Cauchy extension operator E=L◦F on SΩ. To begin with, we need a restriction in the indices so that Eis well-defined from Bp,q sinto Hol(TΩ). As in the case of [2], we will show that for 1 < p, q < +∞this can happen if and only if the distribution F−11Ωe−(e|·)belongs to the dual Besov space (Bp,q s)∗≃Bp0,q0 −(q0−1)s, which is equivalent to say that q < Qs(p). = min j=1,...,r sj+d 2(r−j) d 2(r−j)−n rp + . Under these assumptions on p,qand s, we can prove the next result, which embodies the concept of boundary values. Besov Spaces and Bergman Projections 25 Here the first two statements generalize Theorem 1.7 in [2] to the family of ∆sweights. The last part deserves greater relevance, since it does not have a counterpart even when all indices are equal. This result makes a fundamental use of the continuous notation for Besov spaces (see Corollary 4.7 for the proof). Theorem 1.2. For every Fin Ap,q sthere exists a unique distribution F0∈Bp,q ssuch that F=E(F0). Moreover, 1) limΩ3y→0Fy=F0both in norm of Bp,q sand in S0(V). 2) There exists C > 0s.t. kF0kBp,q s≤CkFkAp,q sfor every F∈ Ap,q s. 3) If 06≡ b ψ∈C∞ c(Ω) is everywhere non-negative, there exists a(ψ)>0 s.t. (1.5) F0=a(ψ)−1ZT∗ Fτ∗−1e∗ψτdτ distributionally, for every F∈ Ap,q s. We can sloppily state Theorem 1.2 by saying that E−1:Ap,q s→Bp,q s is a one-to-one bounded operator, and we shall show in Corollary 4.7 that it also has a dense image. Therefore, two questions may naturally be raised: i) When is E−1an isomorphism? ii) Is there a choice of ψs.t. E−1can be extended to a bounded operator ωsdefined on all of Lp,q s, with Ps=E ◦ ωs? Note that i) holds iff E(Bp,q s)⊆ Ap,q siff E:Bp,q s→ Ap,q sis bounded iff E is an isomorphism from Bp,q sonto Ap,q s. Moreover, we shall show that i) is in a sense equivalent to the statement of ii) w.r.t. the dual indices p0,q0, under some additional assumptions. Indeed, a limiting argument exploiting the arbitrariness of ψin (1.5) will allow to define ωson the core B2,2 sas the (Hilbert) adjoint of E, so that in Corollary 5.2 we will be able to prove that ωsessentially is the (Banach-wise) dual operator of Efor general p,qand s. Finally, self-adjointness of Psand the relation Ps=E ◦ ωs 26 D. Debertol will provide a proof of the following result, where we let qs(p). = min{p, p0}min j=1,...,r 1 + sj−(j−1)d 2 d 2(r−j)!, ps . = 1 + min j=1,...,r sj+n r (r−j)d 2−sj+ . Theorem 1.3. Let sj>(j−1) n r−1 r−1for every j= 1,...,r, and assume that 1< p < ps,q0 s(p)< q < Qs(p). Then, the following properties are equivalent: 1) Psadmits a bounded extension from Lp,q sonto Ap,q s. 2) Eis an isomorphism from Bp,q sonto Ap,q s. We shall have occasion to discuss the significance of the indices qs(p) and pslater, at least in the special case of rank two, that is, for n-dimensional forward light cones. About their relation to this and other questions for higher values of r(but in case s= (ν,...,ν)), corresponding to different levels of difficulty in the original problem, we suggest reading the survey paper [5]. Theorem 1.3 states quite clearly that the existence of bounded extensions of Psinto Lp,q sis strictly related to the characterization of boundary values for functions in Ap,q sas distributions in Bp,q s. Therefore, we shall investigate necessary and sufficient conditions for i) to hold, i.e., such that the operator Ebe an isomorphism from Bp,q sonto Ap,q s. To this aim, the better contribution to proving positive results comes from Littlewood-Paley inequalities as in Lemma 4.8 of [2] (see the proof of Theorem 4.8), which is where the stronger restriction imposed by qs(p) is needed. When sis constant, this produces sharp results for 1 ≤p≤2, see [3], [2], while more sophisticated techniques must be used for p > 2 (see Section 5 in [2]). A main difference in this paper is that, when sis non-constant, the sufficient conditions of Corollary 1.4 below are no longer known to be sharp even for p= 2 unless r= 2 (see Corollary 5.9). Necessary conditions originating from two different ways of producing counterexamples are summarized in Corollary 4.10. Note that the index eqs(p) occurring there is not smaller than qs(p) (and generally strictly bigger). Besov Spaces and Bergman Projections 27 Conclusions about Psare gathered below. When p=q, some weaker results than those presented here for tubes TΩwere obtained in [4], but for a more general class of domains. Corollary 1.4. Let sj>(j−1) n r−1 r−1for every j= 1,...,r, and assume that 1< p < ps,q0 s(p)< q < Qs(p). Then, the following facts are true: 1) Psis bounded on Lp,q sif q < qs(p). 2) Psis unbounded on Lp,q sif p≤p0 sor if q > eqs(p), and even when q=eqs(2) if p≥2. As (5.4) and (5.12) show, the conditions q < Qs(p) and p < pstrivially have to hold if Psis bounded. Note that psplays no role in case s= (ν,...,ν), for then ν > n r−1 is automatically satisfied, and ps= +∞in this case. This also means that the case studied here encompasses the one treated in [2], at least when r > 2: in fact, for the special case of light cones, the authors of [2] can use recent progress on the cone multiplier problem (see e.g. [14]) to obtain sharp results when pis sufficiently large. In the final part of the paper we briefly address the situation in the rank two case, pointing to the gap left open between positive and negative results when s16=s2. Finally, I want to express my appreciation to Professor Fulvio Ricci for his advice and warm encouragement throughout the preparation of this work. I also wish to thank the referee for many suggestions which helped to improve the presentation of this material. 2. Preliminaries 2.1. Notations and basic facts. We introduce the notation and a list of technical results on symmetric cones, mostly taken from [7], [2]. As a guideline, the reader might think of Ω as the cone of r×rreal and positive-definite symmetric matrices. •Ω is an irreducible symmetric cone inside a vector space Vwith inner product (· | ·) and real dimension n. •Vis endowed with a Jordan algebra structure with identity esuch that (x|y) = tr(xy). •The rank rof Ω is the cardinality of any Jordan frame of V. We fix a Jordan frame (c1,...,cr) through the rest of the paper. 28 D. Debertol •V=L1≤i≤j≤rVij is the Pierce decomposition associated with the Jordan frame (c1,...,cr), with Vii = sp{ci}and dimRVij =dif i < j, so that n r−1 = d(r−1) 2. •∆1,...,∆rare the principal minors of Vw.r.t. the fixed Jordan frame. ∆jis a homogeneous polynomial of degree j. ∆r≡∆ is the determinant function. •∆s . = ∆s1 1(∆2/∆1)s2·...·(∆/∆r−1)sris a generalized power on Ω, for s∈Rr. •Gis the identity component of the group {g∈Gl(V) : g(Ω) = Ω}. •G=NAK is the Iwasawa decomposition. •K=G∩O(V) is the stabilizer of ein G. •T=NA is the triangular subgroup associated to the Pierce decomposition of V. •Tacts simply transitively on Ω. That is, the map T3t7→ t·e∈Ω is a diffeomorphism (Gauss decomposition). •If Pdenotes the quadratic representation of V, we have A={Pa| a=Pr j=1 ajcj∈Ω}, and ∆s((nPa)·x) = a2s1 1·...·a2sr r∆s(x),∀n∈N, Pa∈A. •∆ is also invariant under K. Moreover, (2.1) Det g= ∆(g·e)n r,∀g∈G. •The G-invariant measure for Ω is given by (2.2) meas(B). =ZB dy ∆(y)n r . •We say that s>tif and only if sand tboth belong to Rrand they satisfy sj> tjfor each j∈ {1,...,r}. Similarly for s≥t. •g0is the r-tuple whose j-th component is d 2(j−1) = (j−1) n r−1 r−1. •The generalized Gamma function is defined for s∈Crand y∈Ω by ΓΩ(s;y). =ZΩ e−(ξ|y)∆s(ξ)dξ ∆(ξ)n r . The integral is absolutely convergent if and only if <es>g0, and in this case (2.3) ΓΩ(s;y) = ΓΩ(s)∆s(y−1), where we let ΓΩ(s;e). = ΓΩ(s). Besov Spaces and Bergman Projections 29 •For y∈Ω, we have ∆s(y−1) = ∆∗ −s∗(y), where ∆∗ jdenotes the j-th principal minor w.r.t. the rotated Jordan frame (cr,...,c1) and s∗. = (sr,...,s1). •dis the Riemannian G-invariant distance on Ω whose associated metric agrees with (·|·) on the tangent space at Vin e.B(ξ, δ) is the d-ball of radius δcentered at ξ∈Ω. Finally, if A, B > 0, we write ABmeaning that A≤C()B;A∼B stands for both ABand BA. Also, we say that a function f: Ω →R+is locally almost constant if f(ξ)∼δf(η) whenever d(ξ, η)δ, so that we can quote the following result: Lemma 2.1 ([2, 2.4, 2.9]).The principal minors are locally almost constant. The same is true for the functions (· | y)on Ω, uniformly for y∈Ω. 2.2. The Bergman spaces Ap,q s. We define the tube over Ω in the complexification VCof Vas follows: TΩ. =V+iΩ⊂VC. Definition 2.2. For p, q ∈[1,+∞] and s∈Rr, let Lp,q sdenote the (Banach) space of measurable functions on TΩsuch that kfkLp,q s . = ZΩZV |F(x+iy)|pdxq p ∆s(y)dy ∆(y)n r!1 q <+∞, and define the Bergman space Ap,q sas the subspace of Lp,q sformed by its holomorphic functions, Ap,q s . =Lp,q s∩ Hol(TΩ). Of course, dx and dy are the usual Lebesgue measure on V, so that ∆(y)−n rdy is the “left Haar measure” on Ω, as already noted in (2.2). The choice of ∆sentails that a particular Jordan frame (c1,...,cr) has been fixed, once and for all. Note that Ap,∞ sis the Hardy space Hpover the tube TΩ, independently of the choice of s. As a matter of notation, we will name elements of Ap,q sby capital letters, such as F, and by Fytheir sections in Lp(V, dx) at fixed height y∈Ω. 36 D. Debertol We shall see in a moment that in fact, if q6= +∞, the condition on s is also sufficient. A stronger result, which can be adapted from the proof of Corollary 4.5 in [3], is the following: Proposition 2.14. Let s,tin Rrbe strictly bigger than g0and q6= +∞. Then, Au,v t∩ Ap,q sis dense in Ap,q s. On the other hand, it is also possible to give a direct proof of the characterization of the non-triviality of Ap,q s, a result which already is in [3] for light-cones, with the additional assumption s= (ν,...,ν). Theorem 2.15. Let 1≤q < +∞,1≤p≤+∞and s∈Rr. Then, Ap,q sis non-trivial if and only if s>g0. Proof: The necessary condition has already been proved in Proposition 2.13, while for the sufficiency part it is enough to show that the function FN(z). = ∆(e−iz)−Nbelongs to Ap,q sif Nis chosen sufficiently big. But the inequality k(FN)ykp∆(y+e)n rp −N holds, by Corollary 2.12 in case p < +∞and by [10, 7.5] otherwise. Then, the result follows from (2.12) as soon as we choose N > 2n r−1. Therefore, in the sequel we will be justified in assuming s>g0if necessary, so that all of the previous results hold without any further restriction. 3. Besov spaces for Ω The aim of this part of the paper is to define a continuous version of the Besov-type spaces of distributions adapted to the geometry of the symmetric cone Ω, introduced in [2]. These spaces will play a crucial role later on, concerning the description of boundary values and duals of Bergman spaces, and above all in connection with the boundedness of the Bergman projector. Moreover, the new formulation can be best employed in connection with the study of multipliers between (possibly different) Besov spaces, providing a unified approach to this part of the theory, see e.g. Theorem 3.17. An important remark about notations: principal minors ∆kand the triangular group Tare defined w.r.t. the same Jordan frame (c1,...,cr) fixed before. Then, the group T∗made of the adjoints of the elements of Tis the triangular subgroup w.r.t. the rotated Jordan frame (cr,...,c1), and we shall tacitly identify T∗with Ω by means of the diffeomorphism T∗3τ7−→ τe∈Ω. Besov Spaces and Bergman Projections 37 3.1. Preliminaries. Definition 3.1. For a subset Cof V, we call SC(respectively, DC) the set of Schwartz functions on Vwhose Fourier transform is supported (respectively, compactly supported) in C. Now, if ψbelongs to DΩ, define for every τ∈T∗ (3.1) ψτ. =1 (2π)nF−1(b ψ◦τ−1). Clearly, we have that the L1norm of ψτdoes not depend on τ, and we will indifferently denote ψτby ψξif ξ=τe. Also, we will let dτ stand for the left Haar measure on T∗obtained pulling back the measure (2.2) on Ω by means of the natural identification above. Then, the continuous decomposition result we alluded to before is the next one: Proposition 3.2. Assume that (3.2) ZT∗b ψτ(e)dτ = 1. Then, every fin SΩhas the representation f=ZT∗ f∗ψτdτ as a convergent integral in the Fr´echet space S(V). Proof: It depends on a lemma giving the appropriate estimates for f∈ SΩ, extending non-trivially the analogous result in [2, 3.11]. Since a new set of estimates for principal minors is involved, we will sketch the proof below. Lemma 3.3. Let M∈N,t∈Rrwith t1≥t2≥ · · · ≥ tr≥0. Then, there exists L∈N, depending on Mand t, such that for every ξ∈Ω, f∈ SΩand p∈[1,+∞]we have (3.3) kf∗ψξkLp(V,dx)t,M pL(f)∆∗ t(ξ)∆(ξ)n rp0(1 + |ξ|)−M. We are denoting by pLthe Schwartz seminorm pL(η). = sup |α|≤L sup x∈V (1 + |x|)L|∂αη(x)|, η ∈ S(V). 38 D. Debertol Proof: The result was proven in [2] for the case t= (m,...,m), and the (equivalent) inequality (3.4) |b f(ξ)| m,M pL(f)∆m(ξ)(1 + |ξ|)−M was the fundamental step towards the corresponding estimate (3.3), which was then deduced by a standard argument. So, here we will just point out those few addenda we need to adapt the proof of (3.4) to our situation, i.e., to all stated t’s. First of all, we can assume M= 0, since the full statement follows from this particular case by applying it to DMf, if Ddenotes the Laplacian operator on V. Moreover, since b f is supported in Ω, we have that ∂αb fvanishes on ∂Ω, for every α∈Nr. Therefore, choosing las the smallest integer strictly bigger than Pr j=1 tj, and letting dist(ξ, ∂Ω) . = inf σ∈∂Ω|ξ−σ|(ξ∈Ω), we can approximate b f(ξ) with its Taylor polynomial of degree land find |b f(ξ)| lpl(b f) min{dist(ξ, ∂Ω)l,1}max{dist(ξ, ∂Ω)l−1,1}. Thus, we are done if we prove the estimates dist(ξ, ∂Ω) ≤∆∗ k(ξ)1 k for every ξin Ω and kin {1,...,r}, which in turn rest upon the following two facts: 1) dist(Pae, ∂Ω) = minja2 j, if a=Pr j=1 ajcjfor strictly positive aj’s. 2) dist(n∗Pae, ∂Ω) ≤dist(Pae, ∂Ω), for every nin N. Indeed, assume these two facts and write ξ=n∗Pae∈Ω, for some n∈N and Pa∈A, and let ibe such that ai≤ajfor every j. Then, dist(ξ, ∂Ω) ≤a2 i≤(a2 r−k+1 ···a2 r)1 k= ∆∗ k(Pae)1 k= ∆∗ k(ξ)1 k, as claimed. So, we show 1): since a2−a2 jcjbelongs to ∂Ω, the inequality dist(Pae, ∂Ω) ≤ |a2−(a2−a2 jcj)|=a2 j is trivial, for every j= 1,...,r. On the other hand, for any σ∈∂Ω there must exist a primitive idempotent e dwith σe d= 0, by spectral decomposition, and consequently |a2−σ|= sup{|(a2−σ|y)|:|y| ≤ 1} ≥ (a2|e d) = (La2e d|e d); but La2is a positive operator with eigenvalues in na2 h+a2 k 2|h, k=1,...,ro, so that we have shown |a2−σ| ≥ minja2 j. Since this holds for every σ∈∂Ω, 1) is done. We finally prove 2): by 1), it is enough to show that n∗a2−a2 jcjbelongs to ∂Ω for every j∈ {1,...,r}. Since Anormalizes Besov Spaces and Bergman Projections 39 N∗and a2 jcj=Pacj, it is even sufficient (and equivalent) to show that e−ϑcj∈∂Ω for every ϑ∈N∗. But ϑc1=c1and (ϑcj|cj)cj= Pjj (ϑcj) = (Pjj ϑPjj )cj=cj, by the very definition of N∗. Therefore, e−ϑcj= (e−c1) + ϑ(c1−cj)∈Ω and (Le−ϑcjcj|cj) = 1 −(ϑcj|cj) = 0, so that e−ϑcjcannot belong to the open self-dual cone Ω, showing our claim and thus also concluding the proof of the lemma. Finally, we want to point out that, in effect, a weaker variant of (3.3) will also be useful for us: if uis in Rrand u1≥u2≥ · · · ≥ ur≥0, then (3.5) kf∗ψξkLp(V,dx)t,upL(b f)∆∗ t(ξ)∆∗ −u(ξ+e)∆(ξ)n rp0. That (3.5) hold may be seen by noting that |ξ+e| ≤ r(1+|ξ|) for ξ∈Ω, and that (3.6) ∆∗ k(ξ) |ξ|k on Ω by homogeneity, for any k∈ {1,...,r}. Now we can go back to the proof of Proposition 3.2: first of all, notice that ZT∗b ψτ(ξ)dτ = 1Ω(ξ) on V, by left invariance of dτ. Therefore, since F−1is a continuous operator from S(V) into itself and b f≡0 outside Ω, it is sufficient to show that RT∗f∗ψτdτ exists as a convergent integral in the Fr´echet space S(V). This requires an argument about vector-valued integrals, which is new compared to [2]: let us call Blthe completion of S(V) w.r.t. the norm pl, for every l∈N. Note that Blis actually made up of concrete functions, and it is separable. Therefore, by Pettis Theorem in [19, V.4], continuity of the map τ∈T∗Fl 7−→ f∗ψτ∈Bl is sufficient to perform the Bochner integral RT∗f∗ψτdτ into the Banach space Bl, provided that (3.7) ZT∗ pl(\ f∗ψτ)dτ < +∞. 40 D. Debertol So we are left to show (3.7), for every lin N: but by Hausdorff-Young inequality, equation (2.1) and Lemma 3.3, we have that ZT∗ pl(\ f∗ψτ)dτ l,m,u pe l(f)ZΩ ∆(ξ)m ∆(ξ+e)u dξ ∆(ξ)n r , which is a finite quantity by (2.12) if mand uare chosen sufficiently large. Thus, RT∗f∗ψτdτ exists in every Bl, therefore defining an element of S(V), as claimed. Remark 3.4.Since the modular function ∆T∗of T∗is given by (3.8) ∆T∗(τ) = ∆∗ g∗ 0−g0(τe), see [7, VI.3.9], we have that condition (3.2) may also be stated as ZΩb ψ(ξ)∆∗ 1(ξ)···∆∗ r−1(ξ)−ddξ ∆(ξ)= 1. 3.2. Continuous and discrete descriptions. As mentioned in the introduction, we shall consider the following quantity, which will soon be seen to define a norm on SΩ: (3.9) kfkBp,q s . =ZT∗ ∆s((τe)−1)kf∗ψτkq Lp(V,dx)dτ1 q ; here p,qbelong to [1,+∞] (with the obvious modification if q= +∞), while sis in Rr. Lemma 3.5. The relative Schwartz topology is finer than the k kBp,q stopology on SΩ. Proof: We may assume that qis finite, otherwise the result follows from (3.3) and (3.6). With this proviso, by (3.5) we have that kfkBp,q st,upL(f)ZT∗ ∆∗ qt−s∗(τe)∆∗ −qu(τe+e)∆(τe)nq rp0dτ1 q , for every fin SΩ. Owing to (2.12), the integral on the right is convergent if tand uare chosen (and fixed) sufficiently large. Note that actually (3.9) defines a norm on SΩ, by continuity of the map T∗3τ7→ f∗ψτ∈Lp(V, dx). Definition 3.6. We define the Besov space Bp,q sas the completion of SΩ w.r.t. the norm (3.9). Our intention is now to produce a concrete realization of Bp,q sas a subspace (in fact, a quotient subspace) of distributions with spectrum in Ω. Besov Spaces and Bergman Projections 41 Definition 3.7. For a closed subset Cof V, we define S0 Cas the set of tempered distributions on Vwhose Fourier transform is supported in C. From now on we adopt the convention to replace any occurrence of L∞ by C0. Definition 3.8. For p, q ∈[1,+∞] and s∈Rr, let Lp,q s(T∗) denote the vector-valued Lebesgue space Lq(T∗,∆∗ −s∗dτ) taking its values in Lp(V, dx). Then, we define e Bp,q s . ={S∈ S0 Ω|(τ7−→ S∗ψτ)∈Lp,q s(T∗)}/S0 ∂Ω. As a matter of notation, we let hU, hisimply stand for U(h) and ˇ h(x). =h(−x) for functions h∈ S(V) and distributions U∈ S0(V). We first claim that for S∈ S0 Ωwe have S∗ψτ= 0 for almost every τin T∗⇐⇒ S∈ S0 ∂Ω, where we are still assuming that (3.2) holds (actually, it is even sufficient to assume RT∗b ψτ(e)dτ 6= 0). Indeed, if gis a Schwartz function whose support is contained in Ω, by Proposition 3.2 we have that (3.10) hb S, gi=ˇ S, ZT∗bˇg∗ψτdτ=ZT∗ hS∗ψτ,bgidτ. Therefore, independently of the representative chosen for [S], we can define k[S]ke Bp,q s . =k(τ7−→ S∗ψτ)kLp,q s(T∗). Remark 3.9.We show now that any other Schwartz function θin DΩ satisfying c(θ). =RT∗b θτ(e)dτ 6= 0 gives raise to an equivalent norm, so that in particular the finiteness of k k e Bp,q sdoes not depend on the function ψchosen, and consequently e Bp,q sis intrinsecally determined as a set. Proof: It clearly suffices to prove just a one-sided inequality, that in a somewhat sloppy but hopingly self-explanatory notation we write as (3.11) k kθ k kψ. As far as (3.11) is concerned, it is not even necessary to assume that c(θ)6= 0. It is easily seen that (3.10) implies that we have the following weak∗ decomposition for tempered distribution Swhose Fourier transform is compactly supported in Ω: S=ZT∗ S∗ψτdτ. 42 D. Debertol Now, assume that the function τ7→ S∗ψτis in Lp,q s(T∗): in particular, for a fixed τ0∈T∗we have (3.12) kS∗θτ0kpZB(τ0,2R) kS∗ψτkpdτ, if Ris chosen in such a way that both supports of b ψand b θare contained in B(e, R). Thus, if q= +∞we conclude easily by left invariance of dτ. Otherwise, we can apply H¨older inequality, and integrating both sides in ∆∗ −s∗dτ0over T∗we finally end up with (3.11). In order to conclude that e Bp,q sis indeed a realization of Bp,q s, it is convenient to introduce a discrete version of these Besov spaces, patterned after the presentation given in [2]. We recall the basic facts we need. The fundamental tool is a covering lemma, the Whitney decomposition of the symmetric cone Ω, in the form given in [2, 2.6]. It asserts that to any fixed δ > 0 one can associate a sequence of points (ξj)j∈Nand a partition {Ej|j∈N}of Ω in such a way that B(ξj, δ/2) ⊆Ej⊆B(ξj, δ) and that, moreover, the collection of balls {B(ξj, R)|j∈N}of any radius R > 0 has the finite intersection property. Then, proceeding as in [2, 3.2], one can construct a family of C∞functions bχjsubordinated to this covering (with δ. = 1, say) whose inverse Fourier transforms are uniformly bounded in L1(V, dx) and such that Pj∈N|bχj|2is bounded below. Definition 3.10. Given a distribution S∈ S0(V), let kSk¨ Bp,q s . = X j∈N ∆s(ξ−1 j)kf∗χjkq Lp(V,dx)  1 q , with the usual modification in case q= +∞. Then, the discrete Besov space ¨ Bp,q sis defined as ¨ Bp,q s . ={S∈ S0 Ω| kSk¨ Bp,q s<+∞}/S0 ∂Ω. Following closely the footsteps in [2] and mimicking the proofs therein, one can show without much effort that the statement below holds in this generalized setting, in the full range of p,qand s. Besov Spaces and Bergman Projections 43 Proposition 3.11 ([2, 3.25]).Let p, q ∈[1,+∞]and s∈Rr. Then, ¨ Bp,q sis a Banach space, with DΩas a dense subspace. If, in addition, Pjbχj= 1Ω, then (3.13) [S] = X j∈N [S∗χj] for every [S]∈¨ Bp,q s. In the end, to round things off, we are left to prove that the continuous version is equivalent with the discrete one. More precisely, choose any function ψ∈ DΩsatisfying c(ψ)6= 0, and let (ξj)j∈Nbe any δ-lattice, δ > 0, for which the following condition holds: there exists a strictly positive number msuch that (3.14) Φ(ξ). =X j∈N |b ψξj(ξ)|2≥m for ξ∈Ω. Then, the claim is that for S∈ S0 Ωwe have (3.15) k[S]ke Bp,q s∼ψk[S]k¨ Bp,q s, so that we can finally state: Proposition 3.12. e Bp,q s=¨ Bp,q s. Indeed, by (3.15) we would deduce that the identity mapping of e Bp,q s into ¨ Bp,q sis bicontinuous, and we already know the latter to have a complete norm. Incidentally, (3.15) and (3.11) also show that the choice of different Whitney lattices, even w.r.t. different δ’s, does not affect the structure of ¨ Bp,q sas a topological vector space, but it only leads to equivalent norms. Proof of (3.15):The inequality in the ψdirection is easier, and in this case (3.14) plays no role: if [S] is in e Bp,q s, (3.12) tells us that S∗ψξjis in Lp(V, dx) (respectively, in C0if p= +∞) for every j∈N, and that for sufficiently big Rwe have kS∗ψξjkpZB(ξj,2R) kS∗ψτkpdτ. Therefore, for q= +∞the result follows trivially from left invariance of dτ. Otherwise, applying H¨older inequality, Lemma 2.1 and left invariance again, we find ∆s(ξ−1 j)kS∗ψξjkq pZB(ξj,2R) ∆∗ −s∗(τe)kS∗ψτkq pdτ. 44 D. Debertol Summing over j, we conclude by the finite intersection property of Whitney lattices. Now, for the opposite inequality: note that the function Φ belongs to C∞(Ω) and it is bounded below and above, by (3.14) and the finite intersection property of a Whitney lattice. Therefore, letting Nτ. ={k∈N|B(ξk, R)∩B(τe, R)6=∅} for τ∈T∗, we can write S∗ψτ= (2π)−nX k∈Nτ (S∗ψξk)∗ˇ ψξk∗ F−1 b ψτ Φ!∈Lp(V, dx) with (3.16) kS∗ψτkpX k∈Nτ kS∗ψξkkp. Moreover, the sets Nτare locally almost constant, and as a consequence the function τ∈T∗7→ S∗ψτis continuous. So, for q= +∞we conclude easily by (3.16), and otherwise carrying on as usual we find that for τ in Ej kS∗ψτkq pX k∈Nj kS∗ψξkkq p. Finally, using Lemma 2.1 once more and interchanging a summation order, we obtain k[S]kq e Bp,q s X j∈N ∆s(ξ−1 j)ZEj kS∗ψτkq pdτ X j∈NX k∈Nj ∆s(ξ−1 k)kS∗ψξkkq p k[S]kq ¨ Bp,q s. 3.3. Invariance and duality. Therefore, from now on we can and shall feel free not to distinguish between these two equivalent realizations of the Besov space Bp,q s, even if we will usually adopt the continuous version as being the most convenient one. For instance, the proposition below, which is in [2, 3.8] for the discrete setting, becomes trivial here. Note that, consistently with the idea to represent Bergman spaces by means of Besov spaces, the norm (3.9) had to be defined that way, and w.r.t. T∗, in order to match exactly the same homogeneity relation w.r.t. Tshown in (2.4). Besov Spaces and Bergman Projections 45 Proposition 3.13. Bp,q sis invariant for the action of T, and (3.17) k[S]◦tkBp,q s=k[S]kBp,q s∆s(te)−1 q∆(te)−n rp . On the other hand, it is easier to determine the dual of a Besov space by working with its discrete version. Since the general case consists in a plain adaptation of the corresponding result [2, 3.27], we omit the proof of the next statement. Proposition 3.14. Let p,qbelong to ]1,+∞[and sto Rr. Then, the dual space of Bp,q sand Bp0,q0 −(q0−1)sare isomorphic, with a duality pairing given by (3.18) Bp0,q0 −(q0−1)s h[U],[S]iBp,q s . =X j∈N hU∗eχj,(S∗χj)ˇi which is independent of the representatives chosen for [U],[S]and of the functions χj,eχj(here, in addition to the requirements made on both families (bχj) and (beχj) prior to Definition 3.10, we are also assuming Pjbχj= 1Ωand beχjbχj=bχj). Note however that (3.18) can also be given a continuous interpretation, for which it is sufficient to consider any pair of functions χ, eχ∈ DΩ such that c(χ) = 1, c(eχ)6= 0 and beχbχ=bχ. Indeed, simply let (3.19) Bp0,q0 −(q0−1)s h[U],[S]iBp,q s . =ZT∗ hU∗eχτ,(S∗χτ)ˇidτ, and note that the two formulas agree (with (2π)−nhb U, b fi) if S. =fis in the dense subspace DΩ, and that for both the estimate (3.20) |h[U],[S]i| ≤ k[U]kBp0,q0 −(q0−1)s k[S]kBp,q s holds for arbitrary [U], [S], by applying H¨older inequality twice. 3.4. Multipliers. Definition 3.15. For pin [1,+∞], we let Mpdenote the Banach space of Lp(V, dx) multipliers with the induced norm as a subspace of the bounded linear operators from Lp(V, dx) into itself. Mpbeing a non-separable space (it contains the uncountable, discrete subset formed by translation operators), weak measurability of an Mp-valued function is no longer sufficient for implementing a Bochner integral. Therefore, we need the following (standard) assumption, see e.g. [19]: 52 D. Debertol Lemma 4.4. Assume that p,qbelong to ]1,+∞[,sis in Rrand that condition (4.1) holds. Then, for every [S]in Bp,q s, the integral (4.6) ZT∗ S∗ψτdτ distributionally defines an element S]in the equivalence class of [S]. Moreover, the discrete decomposition (4.7) S]=X j S∗χj holds and the limit (4.8) lim Ω3y→0(E[S])y=S] exists, both with convergence in S0(V). In particular, S]is independent of the representative chosen for [S]and Eis injective on Bp,q s. Note that (4.6) and (4.7) do not depend on the choice of ψor χj’s provided they satisfy c(ψ) = 1 and Pjbχj= 1Ω. Proof: Formula (4.6) simply means that for gin S(V) we let hS], gi. =ZT∗ hS∗ψτ, gidτ. Now, if θ∈ DΩis such that c(θ)6= 0 and b θb ψ=b ψ, by applying H¨older inequality twice as in (3.20) we obtain |hS], gi| ≤ ZT∗ |hS∗ψτ,(ˇg∗θτ)ˇi| dτ  k[S]kBp,q sk(τ7−→ ˇg∗θτ)k(Lp,q s(T∗))∗, (4.9) and the right-hand side is a finite quantity for any gin S(V) because of conditions (4.1) and Lemma 4.2. This is sufficient to conclude that S]is in S0(V), and then easy manipulations with (4.6) show that it also belongs to S0 Ω. Moreover, if his a Schwartz function supported in Ω, we can apply Proposition 3.2 to bˇ hobtaining that hc S], hi=ZT∗ hS∗ψτ,bhidτ =ˇ S, ZT∗bˇ h∗ψτdτ=hb S, hi, or equivalently [S]] = [S]. Next, note that the estimate (4.10) *X j S∗χj, g+ k[S]kBp,q sk(ˇg∗χj)jk(`p,q s)∗ Besov Spaces and Bergman Projections 53 holds as well for every g∈ S(V), by a trivial adaptation of the proofs of (4.9) and Lemma 4.2 to the discrete setting. Therefore, the map [S]∈Bp,q s7−→ S]−X j S∗χj∈ S0(V) is continuous, and it clearly vanishes on DΩ, so that (4.7) is established by density. Finally, the last part of the statement follows from Proposition 4.1 once we note that [(E[S])y]]= (E[S])y in S0(V), for every [S]∈Bp,q sand y∈Ω. But (E[S])y=Pj∈N(E[S∗χj])y in C0(V) and therefore in S0(V), by (3.13), (4.2) and the continuity of E. Hence, it is enough to show (4.11) (E[S∗χj])y= (E[S])y∗χj for every j∈N, and in turn this last equality is a consequence of (4.5). Remark 4.5.Note that as a by-product of (4.7) and Remark 4.3, under the assumptions made in (4.1) one can also extend the validity of (4.12) E[S] = (2π)−nLc S] to every [S] in Bp,q s, for Lis continuous on S0(V), see [13, VII.4.2]. From now on, condition (4.1) and its dual (4.3) will appear in an increasing number of situations, each time assuming the role of critical indices for some property. This is indeed the case in Proposition 4.1 and in Lemma 4.2, which do not hold beyond these indices (at least for p and qstrictly between 1 and +∞). The proof of these two observations is a simple modification of the one presented in [2, 3.48], and so we skip it. 4.2. The proof of Theorem 1.2. We now prove a partial result in the direction of representing a Bergman space by means of the Cauchy operator defined on the corresponding Besov space, extending [2, 1.7]. The proof is slightly simpler though, for the continuous notation is employed. Theorem 4.6. Let pand qbelong to ]1,+∞[,sto Rrand assume that condition (4.1) holds. Then, E(Bp,q s)contains Ap,q sand the operator E−1 defined from Ap,q sinto Bp,q sis bounded with dense image. In particular, (4.13) lim Ω3y→0k[Fy]− E−1FkBp,q s= 0 for every Fin Ap,q s. 54 D. Debertol Proof: By Theorem 2.15 we may also assume that s>g0, for else there is nothing to prove. In this case, we know from Proposition 2.14 that Ap,q s∩H2is a dense subspace of Ap,q s. Therefore, to any F∈ Ap,q s∩H2we can associate bgin L2(Ω, dξ) such that F= (2π)−nLbg, by Theorem 2.7. Moreover, g=Pj∈Ng∗χjin L2(V, dx) (and consequently in S0(V)) by Plancherel formula and dominated convergence. Thus, if we can show that gis in Bp,q s, we also find that F=E[g], by (4.12). But kg∗ψτkp kF−1(bge−(τ∗−1e|·))kpkF1(b ψe(e|·))k1 =kF(τe)−1kpkF−1(b ψe(e|·))k1. Therefore, k[g]kq Bp,q sZT∗ kF(τe)−1kq p∆s((τe)−1)dτ =ZΩ kFξ−1kq p∆s(ξ−1)dξ ∆(ξ)n r =kFkq Ap,q s, (4.14) for the invariant measure (2.2) is preserved under inversion, see [7, II.3.3]. In the general case we may proceed by density. Finally, we show that the image of E−1contains DΩ. So, pick up gin DΩand let Hdenote the support of bgand b ϑbe any function in C∞ c(Ω) which is identically 1 on H. Then, there exists γH>0 such that k(Eg)ykp kg∗ F−1(e−(y|·)b ϑ)kpHkgkpe−γH(y|e) independently of yin Ω. Since s>g0, we can easily conclude by (2.3) that kEgkAp,q sis finite. In fact, Lemma 4.4 is more precise than Theorem 4.6 as to (4.13), since in addition it asserts that the limit of Fyexists in S0(V) as well, for ytending to 0 inside Ω and Fin Ap,q s. What we can still do in the hypothesis of Theorem 4.6 is to explicitly determine this limit, that is, (E−1F)]. This part is new if compared to [2], and it gives the “correct” definition for the operator E−1, according to Lemma 4.4. Heuristically, the idea comes from “discretizing” (4.12) with b S=bg in L2(Ω, dξ) at height yand then let ytend to 0: Fy=X j∈NZB(ξj,2) bg(ξ)bχj(ξ)ei(·+iy|ξ)dξ ∼X j∈N e−(y|ξj)(Fξ−1 j∗χj). Besov Spaces and Bergman Projections 55 However, in order to make this argument work, we must introduce an appropriate invariance property which is most conveniently exploited in the continuous notation. In detail, once a function ψin DΩ\{0}whose Fourier transform is everywhere non-negative on Ω has been fixed, define a(ψ). =ZΩ e−(ξ|e)b ψ(ξ)∆∗ 1(ξ)···∆∗ r−1(ξ)−ddξ ∆(ξ), which is a strictly positive number, and for Fin Ap,q slet (4.15) hF0, hi=a(ψ)−1ZT∗ hFτ∗−1e∗ψτ, hidτ (h∈ S(V)). Then, we claim that F0is a well defined distribution in S0 Ωand the canonic representative of E−1Fin Bp,q sat the same time. First of all, proceeding as in (4.9) we obtain the inequality (4.16) |hF0, hi|ZΩ kFξ−1kq p∆s(ξ−1)dξ ∆(ξ)n r1 q k(τ7−→ˇ h∗ψτ)k(Lp,q s(T∗))∗, and therefore (4.14) and Lemma 4.2 show that F0is a tempered distribution whose Fourier transform is supported in Ω. Then, note that there exists R > 0 such that for every η∈T∗we have kF0∗ψηkq pZB(η,R) kFτ∗−1ekq pdτ. In particular, by Lemma 2.1 and invariance we may estimate k[F0]kq Bp,q sZT∗ ZB(η,R) kFτ∗−1ekq pdτ!∆s((ηe)−1)dη ZT∗ ZB(τ,R) kFτ∗−1ekq pdη!∆s((τe)−1)dτ ZΩ kFξ−1kq p∆s(ξ−1)dξ ∆(ξ)n r , (4.17) which finally is kFkq Ap,q s, by (4.14) once more. The next step consists in proving that [F0]]=F0, or equivalently, that hF0−PN j=0 F0∗χj, hitends to 0 as Ndiverges, for every hin S(V). But *F0− N X j=0 F0∗χj, h+=*F0, h − N X j=0 h∗ˇχj+, 56 D. Debertol so that *F0, h− N X j=0 h∗ˇχj+ kFkAp,q s F−1 1Ω bˇ h− N X j=0 \ ˇ h∗χj   (Bp,q s)∗ by the estimate (4.16), and actually  F−1 1Ω N X j=0 \ ˇ h∗χj  = N X j=0 [ˇ h∗χj] tends to [F−1(1Ωbˇ h)] in (Bp,q s)∗by (3.13) and Proposition 3.14, for k[F−1(1Ωbˇ h)]k(Bp,q s)∗=k(ˇ h∗ψτ)τk(Lp,q s(T∗))∗ is finite by Lemma 4.2. Finally, we show that F0is the limit of the net (Fy)yin S0(V) as yapproaches 0 inside Ω. By Theorem 4.6 and (4.17) we may assume that F=Egfor some bgin L2(Ω, dξ). For hin S(V) we then have (4.18) a(ψ)hc F0, hi=ZT∗ h\ Fτ∗−1e, h b ψτidτ =ZT∗ he−(τ∗−1e|·)bg, h b ψτidτ. But ZT∗ e−(e|τ−1v)b ψ(τ−1v)dτ =a(ψ) independently of vin Ω, by left invariance of dτ and (3.8), so that applying Fubini’s theorem in (4.18) we can conclude that F0=g, as claimed. We record this result in a statement for future reference: Corollary 4.7. Let pand qbelong to ]1,+∞[,s>g0and assume that condition (4.1) holds. Then, the map F∈ Ap,q s7−→ F0∈Bp,q s defines a bounded linear operator with a dense image, and F=E[F0]. Moreover, F0= [F0]]is the limit of Fywhen ytends to 0inside Ω, both in the norm of Bp,q sand in S0(V). 4.3. The isomorphism between Ap,q sand Bp,q s. Corollary 4.7 embodies the fact that a function Fin Ap,q shas a boundary value F0in Bp,q s, and that we can get Fback extending F0to the tube domain over Ω by means of the Cauchy operator E. However, it still does not say if every distribution in Bp,q sis obtained this way, that is, whether E(Bp,q s) properly contains Ap,q sor else, if these Besov Spaces and Bergman Projections 57 two spaces coincide. And in fact, we will see that both possibilities can occur, as [2, 1.9] already shows in case s= (ν . . . , ν). Therefore, we shall prove in a moment a strengthening of Theorem 4.6 stating that E(Bp,q s) = Ap,q s, though under some further restrictions on p,qand s, and afterwards we shall look for counterexamples. Even if some of this material clearly is an adaptation of the corresponding results in [2], we will give the proofs, to illustrate where the additional restrictions related to the weights ∆scome from. Before doing this we restate condition (4.1): under the assumption s> g0it is equivalent with (4.19) q < Qs(p). = min j=1,...,r sj+d 2(r−j) d 2(r−j)−n rp + , where for strictly positive cand arbitrary (real) awe let a+. = max{a, 0}and c 0 . = +∞. Note that the inequality Qs(p)> p0always holds, since s>g0; nevertheless, and unlike the situation in [2], here Qs(p)<2 can occur. At the other extreme, Qs(p) = +∞when p0≥n r, which in particular is always the case if n= 1. We now introduce the index qs(p). Let p]. = min{p, p0}; then, we define (4.20) qs(p). =p]min j=1,...,r 1 + sj−(j−1)d 2 d 2(r−j)!. For s>g0we have that 1 < qs(p)< Qs(p), as it can be seen by comparing at same j’s. Moreover, the requirement q < qs(p) can be equivalently stated in terms of sas (4.21) s>g0+q p] −1+ g∗ 0. In case s= (ν,...,ν), condition (4.20) then simplifies to ν > n r−1 and q < p]1 + ν n r−1, so that the next theorem actually extends the result in [2, 1.8], at least for p, q 6= 1: Theorem 4.8. Let p,q,ssatisfy condition (4.21) for some 1< p, q < +∞. Then, Eis an isomorphism of Bp,q sonto Ap,q s. 58 D. Debertol Proof: By Corollary 4.7 we only need to prove a bound below on the norm of E−1, which amounts to the estimate kEfkAp,q s kfkBp,q s,∀f∈ DΩ. The crucial step is the following elementary lemma: Lemma 4.9 ([2, 4.8]).Let 1≤p≤+∞,1≤v≤p]. Then, the linear operator (fj)j∈N∈`v(Lp(V, dx)) 7−→ X j∈N fj∗χj∈Lp(V, dx) is bounded. Thus, applying the lemma above with fj. =f∗F−1(e−(y|·)b θj), and by Young inequality, we can bound k(Ef)ykp=kF−1(e−(y|·)b f)kp=X k∈N F−1e−(y|·)b fbχkp  X k∈N kf∗χkkp] pkF−1(e−(y|·)b θk)kp] 1!1 p] . Moreover, there exists γ > 0 such that kF−1(e−(y|·)b θk)k1e−γ(y|ξk) independently of yin Ω, and consequently (4.22) kEfkq Ap,q sZΩ X k∈N kf∗χkkp] pe−(y|ξk)!q p] ∆s(y)dy ∆(y)n r . Now, if q≤p]we majorize the `p]/q norm of the integrand in the righthand side of (4.22) with its `1norm, finding as claimed that kEfkq Ap,q sX k∈N kf∗χkkq pZΩ e−(y|ξk)∆s(y)dy ∆(y)n r = ΓΩ(s)kfkq Bp,q s, by (2.3). Otherwise, we set u. =q p], so that u > 1. Then, an application of H¨older inequality in (4.22) yields that kEfkq Ap,q sis bounded above by ZΩ X k∈N kf∗χkkq p∆u t(ξ−1 k)e−(y|ξk) ! X k∈N ∆u0 −t(ξ−1 k)e−(y|ξk) !u u0 ∆s(y)dy ∆(y)n r . The second sum is uniformly controlled by a scalar multiple of ∆u −t(y) for yin Ω as soon as u0t>g∗ 0, by Lemma 2.1 and (2.3); thus, by (2.3) Besov Spaces and Bergman Projections 59 once more we find that kEfkq Ap,q sX k∈N kf∗χkkq p∆u t(ξ−1 k)ZΩ e−(y|ξk)∆u −t(y)∆s(y)dy ∆(y)n r kfkq Bp,q s if also s−ut>g0. Forcing the fulfilment of the two conditions on tis equivalent with the requirement that s−g0>u u0g∗ 0=q p] −1g∗ 0. Finally, the conditions arising from two cases can be summarized in (4.21). 4.4. Counterexamples. We shall now address the following question: when is Theorem 4.8 sharp? i.e., is the condition q < qs(p) necessary for Theorem 4.8 to hold? On behalf of Corollary 4.7, Theorem 4.8 is equivalent with the boundedness of the Cauchy operator from Bp,q sinto Ap,q s. Therefore, we shall assume that sis strictly bigger than g0,pbelongs to ]1,+∞[ , 1< q < Qs(p) and that (4.23) kE[S]kAp,q s k[S]kBp,q s([S]∈Bp,q s). First, arguing as in [2,§4.4], one can easily show that (4.23) implies the existence of a constant Ap, exclusively depending on p, such that the inequality X|aj|21 2≤ApX∆s(ξ−1 j)|aj|q1 q holds for any choice of finite subsets Eof {ξj:|ξj|<1}and of scalars aj. Now it’s just a matter of balancing ajwith ∆s(ξ−1 j): indeed, taking aj to be equal to ∆s(ξ−1 j)1 2−q, we have that  X ξj∈E ∆s(ξ−1 j)2 2−q  1 2−1 q ≤Ap independently of the finite subset E, which for q > 2 is equivalent with ZΩ∩(e−Ω) ∆∗ s∗(ξ)2 q−2dξ ∆(ξ)n r <+∞; 60 D. Debertol by (2.13), this can only happen if s∗>q 2−1g0, and in terms of q, only if (4.24) q < 2 min j=1,...,r 1 + sj (r−j)d 2!. The second technique we employ to produce extra necessary conditions on qis based on some direct calculations which provide explicit counterexamples to the surjectivity of the operator E−1. We will get better results than in (4.24) only for p≤2, which is thus what we assume from now on. The idea is to exhibit particular distributions in Bp,q swhose Cauchy transform is not in Ap,q s, thus contradicting (4.23). In detail, for u∈Rr with u>g0we let b Su . = (2π)ne−(e|·)∆∗ u∆−n r1Ω. First of all, note that b Suis an L1function precisely when u>g0, and that in this case it can also be expressed as the L1series Pj∈Nb Subχj, by dominated convergence. Therefore, Su= ΓΩ(u)∆−u∗(e−i·) on V, and it also is the sum of the series Pj∈NSu∗χjin C0(V); in particular the same equality holds in S0(V). Now, set Fu . = (2π)−nLb Su: by (2.3), we have that Fucoincides with the holomorphic function ΓΩ(u)∆−u∗(e−i·) on TΩ, so that Fu=E[Su] when Subelongs to Bp,q s. The usual estimates being essentially too rough, instead of trying to compute k[Su]kBp,q sstraight by the definition we proceed differently, and wonder if t∗ [Su] belongs to Bp,q s+qtfor some t∈Rrwith t>0. At a formal level, the guess is that we should be concerned with an estimate for kFu+t∗kAp,q s+qt, and we now show that this approach actually works. Indeed, by Corollary 2.12 and (2.12) we know that the function Fu+t∗is in Ap,q s+qtif and only if p(u+t∗)>g0+n r,...,n r and s∗+g0< q u−n rp,..., n rp. (4.25) In this case, by (4.12), Theorem 4.6 and the remarks above we also have that Su+t∗is in Bp,q s+qtand that Su+t∗=E−1Fu+t∗. Moreover, −t∗ [Su+t∗] = [Su], Besov Spaces and Bergman Projections 61 because the relation χj∗−t∗ [Su+t∗] = Su∗χjholds for every j∈N, by (4.11). In particular, [Su] belongs to Bp,q s, since −t∗is an isomorphism. Finally, we can prevent Fufrom belonging to Ap,q sby requiring that k(Fu)ekp= +∞, which on behalf of Corollary 2.12 is equivalent to the condition (4.26) pu6>g0+n r,...,n r. Collecting (4.25) and (4.26) altogether, one can see that the best choice (in order to bound qfrom below) comes from fixing j∈ {1,...,r}and letting uj. =1 pn r+ (j−1)d 2. Up to completing uwith suitably high positive values, the only condition effectively remaining is the j-th one of the second set of inequalities in (4.25), so that in the end we have shown that Theorem 4.8 cannot hold when q > p 1 + sr+1−j (j−1)d 2!. Since this is true for every jin {1,...,r}, we then deduce the necessary condition (4.27) q≤pmin j=1,...,r 1 + sj (r−j)d 2!, which we recall has to hold for 1 < p ≤2. Therefore, we can merge (4.24) and (4.27) into the statement below: Corollary 4.10. Let pin ]1,+∞[,s>g0and 1< q < Qs(p). Then, Theorem 4.8 can hold only if q≤min{p, 2}min j=1,...,r 1 + sj (r−j)d 2!. =eqs(p), and equality can occur only if p < 2. In case s= (ν,...,ν), one can show that the inequality in Corollary 4.10 is strict: q < eqs(p), see [2, 4.34]. It seems reasonable to conjecture that this will also be the case for general s. For the rank 2 case, see Theorem 5.8 below. 68 D. Debertol Theorem 5.6. Let s>g0, and assume that 1<p<ps,q0 s(p)<q <Qs(p). Then, the following properties are equivalent: 1) Psadmits a bounded extension from Lp,q sonto Ap,q s. 2) ωsadmits a bounded extension from Lp0,q0 sonto Bp0,q0 s. 3) Eis an isomorphism from Bp,q sonto Ap,q s. 4) jis an isomorphism from Ap,q sonto (Ap0,q0 s)∗. Proof: We start with a preliminary remark: under the stated hypotheses, Theorem 4.8 applies w.r.t. p0,q0and s, so that (5.13) Eis an isomorphism from Bp0,q0 sonto Ap0,q0 s. 1) ⇒2): As a consequence of Remark 5.4 and (5.13), E−1◦Psis bounded from Lp0,q0 sonto Bp0,q0 s, and by Proposition 5.1 it equals ωs on L2,2 s∩Lp0,q0 s. 2) ⇒3): This is part of Corollary 5.2, since the image of e Eis then clearly contained in Ap,q s; surjectivity of Enow follows from Corollary 4.7. 3) ⇒1): According to Proposition 5.3, ωsis bounded from Lp,q s into Bp,q s. Thus, E ◦ ωsis bounded from Lp,q sinto Ap,q s, and it is equal to Pson L2,2 s∩Lp,q sby Proposition 5.1; surjectivity of Psfinally follows from the density result of Proposition 2.14. 3) ⇒4): It is sufficient to show that jis surjective. So, take Φ ∈ (Ap0,q0 s)∗. By (5.13), Ψ . = Φ ◦ E belongs to (Bp0,q0 s)∗. In particular, owing to Proposition 3.14, there exists [U]∈Bp,q −(q−1)srepresenting Ψ. Then, [X]. =s∗[U]∈Bp,q s, so that F. =2s1+···+sr ΓΩ(s)E[X] belongs to Ap,q s. But E ◦ ωsis the identity on Ap0,q0 s, by Proposition 5.1, so that by the adjunction (5.11) we have that for every G∈Lp0,q0 s Φ(G) = Ψ(E−1G) = Bp0,q0 shE−1G, [U]iBp,q (1−q)s =Bp0,q0 shωsG, [U]iBp,q (1−q)s =2s1+···+sr ΓΩ(s)Bp0,q0 −(q0−1)s hπsG, [X]iBp,q s =Lp0,q0 shG, FiLp,q s =j(F)(G), i.e., j(F) = Φ. Besov Spaces and Bergman Projections 69 4) ⇒3): According to Corollary 4.7, it is enough to prove a uniform estimate kEfkAp,q s kfkBp,q s for f∈ DΩ. But jis an isomorphism, so that (5.14) kEfkAp,q ssup{|j(Ef)(G)|:kGkAp0,q0 s≤1, G ∈ A2,2 s}. Since ωs◦ E is the identity on A2,2 s, owing to (5.11) we obtain that j(Ef)(G) = Bp0,q0 −(q0−1)s hπs(G), fiBp,q s =ΓΩ(s) 2s1+···+srBp0,q0 (1−q0)s h−s∗(E−1G), fiBp,q s ; −s∗being an isomorphism, (5.13) and (5.14) give us the result. Remark 5.7.The implications “1 ⇒2”, “2 ⇒3” and “4 ⇒3” still continue to hold in the wider range 1 <p<+∞, 1 < q0< Qs(p0), for they only make use of the boundedness of E−1on Ap0,q0 s, and owing to Corollary 4.7 and (4.19) this happens in the range stated above. Proof of Corollary 1.4: On behalf of Theorem 5.6, 1) is a consequence of Theorem 4.8, while 2) follows from Remark 5.4 and the counterexamples in Corollary 4.10. 5.1. The rank 2 case. Finally, we briefly discuss the situation depicted by Corollary 1.4 in the special case of forward light cones, which are the prototypes for irreducible symmetric cones of rank two. So, for n≥3 let Ω≡Λn. ={y= (y0, yn)∈Rn−1×R|yn>|y0|}. Then, for an appropriate choice of the Jordan frame (c1, c2) we have ∆1(y) = yn−y1,∆(y) = y2 n− |y0|2. Moreover, the usual assumption s>g0becomes s1>0, s2>n−2 2here. Let us remark once more that sharper results are obtained in [2, 5.11] for light cones when s1=s2=ν. But in fact we shall see below that new interesting phenomena occur precisely when s1≤n−2 2, so that in particular s1and s2are forced to be different. 70 D. Debertol Now, note that the various indices involved in Corollary 1.4 all attain their minima at j= 1, and thus we find ps= 1 + 2s1+n (n−2−2s1)+ =   2(n−1) n−2−2s1 ,if s1<n−2 2 +∞,otherwise , Qs(p) = 1 + 2s1 n−2 1−n (n−2)p+ , qs(p) = p]1 + 2s1 n−2, eqs(p) = max{2, p0} p0qs(p). Moreover, one can check that Qs(p) = q0 s(p)⇔p=pswhen s1<n−2 2, and therefore we can translate the statement of Corollary 1.4 into the following: Theorem 5.8. Let n≥3,Ω. = Λn,1< p, q < +∞and s1>0, s2>n−2 2. Then, 1) Psis bounded on Lp,q s(TΩ)if q0 s(p)< q < qs(p). 2) The gap between positive and negative results is given by the region where p≥2and max{q0 s(p), qs(p)} ≤ q < min{qs(2), Qs(p)} and its dual one. And of course, 3) Psis unbounded on Lp,q sin the remaining regions. Note that qs(p) becomes smaller than 2 for values of pclose to 1 and +∞when 2s1≤n−2. Also, one has that Qs(p)< qs(2) if and only if p > 2n n−2, which is an effective restriction only if s1>n−2 2n. In particular, for p= 2 the following statement holds true without exception: Corollary 5.9. Let n≥3,Ω. = Λn,1<q<+∞and s1>0,s2>n−2 2. Then, Psis bounded on L2,q s(TΩ)if and only if q0 s(2) < q < qs(2). Besov Spaces and Bergman Projections 71 References [1] D. B´ ekoll´ e, A. Bonami and G. Garrig´ os, Littlewood-Paley decompositions related to symmetric cones, IMHOTEP J. Afr. Math. Pures Appl. 3(1) (2000), 11–41. [2] D. B´ ekoll´ e, A. Bonami, G. Garrig´ os and F. Ricci, Littlewood-Paley decompositions related to symmetric cones and Bergman projections in tube domains, Proc. London Math. Soc. (3) 89(2) (2004), 317–360. [3] D. B´ ekoll´ e, A. Bonami, M. M. Peloso and F. Ricci, Boundedness of Bergman projections on tube domains over light cones, Math. Z. 237(1) (2001), 31–59. [4] D. B´ ekoll´ e and A. Temgoua Kagou, Reproducing properties and Lp-estimates for Bergman projections in Siegel domains of type II, Studia Math. 115(3) (1995), 219–239. [5] A. Bonami, Three related problems of Bergman spaces of tube domains over symmetric cones, Harmonic analysis on complex homogeneous domains and Lie groups (Rome, 2001), Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 13(3–4) (2002), 183–197. [6] R. R. Coifman and R. Rochberg, Representation theorems for holomorphic and harmonic functions in Lp, in: “Representation theorems for Hardy spaces”,Ast´erisque 77, Soc. Math. France, Paris, 1980, pp. 11–66. [7] J. Faraut and A. Kor´ anyi,“Analysis on symmetric cones”, Oxford Mathematical Monographs. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1994. [8] M. Frazier, B. Jawerth and G. Weiss,“Littlewood-Paley theory and the study of function spaces”, CBMS Regional Conference Series in Mathematics 79, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1991. [9] J. Garc´ ıa-Cuerva and J. L. Rubio de Francia,“Weighted norm inequalities and related topics”, North-Holland Mathematics Studies 116, Notas de Matem´atica 104, North-Holland Publishing Co., Amsterdam, 1985. [10] G. Garrig´ os, Generalized Hardy spaces on tube domains over cones, Colloq. Math. 90(2) (2001), 213–251. [11] S. G. Gindikin, Analysis in homogeneous domains, (Russian), Uspehi Mat. Nauk 19(4) (118) (1964), 3–92. 72 D. Debertol [12] S. Helgason,“Differential geometry, Lie groups, and symmetric spaces”, Pure and Applied Mathematics 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. [13] L. H¨ ormander,“The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis”, Grundlehren der Mathematischen Wissenschaften 256, Springer-Verlag, Berlin, 1983. [14] I. Laba and T. Wolff, A local smoothing estimate in higher dimensions, Dedicated to the memory of Tom Wolff, J. Anal. Math. 88 (2002), 149–171. [15] F. Ricci and M. Taibleson, Boundary values of harmonic functions in mixed norm spaces and their atomic structure, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10(1) (1983), 1–54. [16] E. M. Stein and G. Weiss,“Introduction to Fourier analysis on Euclidean spaces”, Princeton Mathematical Series 32, Princeton University Press, Princeton, N.J., 1971. [17] H. Triebel,“Theory of function spaces”, Monographs in Mathematics 78, Birkh¨auser Verlag, Basel, 1983. [18] ` E. B. Vinberg, S. G. Gindikin and I. I. Pjatecki˘ ı-ˇ Sapiro, Classification and canonical realization of complex homogeneous bounded domains, (Russian), Trudy Moskov. Mat. Obˇsˇc. 12 (1963), 359–388. [19] K. Yosida,“Functional analysis”, Die Grundlehren der Mathematischen Wissenschaften 123, Academic Press, Inc., New York; Springer-Verlag, Berlin, 1965. Scuola Normale Superiore Piazza dei Cavalieri, 7 56126 Pisa Italy E-mail address:[email protected] Primera versi´o rebuda el 4 de desembre de 2003, darrera versi´o rebuda el 9 de febrer de 2005.