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ATOMIC DECOMPOSITION AND INTERPOLATION VIA THE COMPLEX METHOD FOR MIXED NORM BERGMAN SPACES ON TUBE DOMAINS OVER SYMMETRIC CONES DAVID BEKOLLE, JOCELYN GONESSA AND CYRILLE NANA Abstract. Starting from an adapted Whitney decomposition of tube domains in Cn over irreducible symmetric cones of Rn, we prove an atomic decomposition theorem in mixed norm weighted Bergman spaces on these domains. We also characterize the inter- polation space via the complex method between two mixed norm weighted Bergman spaces. 1. Introduction The context and the notations are those of [13]. Let Ω be an irre- ducible symmetric cone of rank r in a vector space V of dimension n, endowed with an inner product (.|.) for which Ω is self-dual. We recall that Ω induces in V a structure of Euclidean Jordan algebra with identity e such that Ω= {x2 : x ∈ V }. Let {c1, ··· ,cr} be a fixed Jordan frame in V and arXiv:1703.07862v1 [math.CA] 22 Mar 2017 V = ⊕ Vi,j 1≤i≤j≤r be its associated Peirce decomposition of V . We denote by ∆1(x), ··· , ∆r(x) the principal minors of x ∈ V with respect to the fixed Jordan frame {c1, ··· ,cr}. More precisely, ∆k(x), k =1, ··· ,r is the determinant of (k) the projection Pkx of x, in the Jordan subalgebra V = ⊕1≤i≤j≤kVi,j. We have ∆k(x) > 0, k = 1, ··· ,r when x ∈ Ω, and the determinant ∆ of the Jordan algebra is given by ∆ = ∆r. The generalized power 1 2 DAVID BEKOLLE, JOCELYN GONESSA AND CYRILLE NANA function on Ω is defined as s1−s2 s2−s3 sr r ∆s(x) = ∆1 (x)∆2 (x) ··· ∆r (x), x ∈ Ω, s =(s1, ··· ,sr) ∈ C . We adopt the following standard notations: n − 1 n − 1 n = 2(k − 1) r and m = 2(r − k) r . k r − 1 k r − 1 r For s =(s1, ··· ,sr) ∈ R and ρ real, the notation s+ρ will stand for the vector whose coordinates are sk + ρ, k =1, ··· , r. For 1 ≤ p ≤∞ and p, q 1 ≤ q < ∞, let Ls denote the mixed norm Lebesgue space constisting of measurable functions F on TΩ such that 1 q q kF k p, q = kF (· + iy)k ∆s n (y)dy < ∞ Ls p − r ZΩ where 1 p p kF (· + iy)kp = |F (x + iy)| dx ZV (with the obvious modification if p = ∞). The mixed norm weighted p,q p, q Bergman space As is the (closed) subspace of Ls consisting of holo- p,q morphic functions. Following [11], As is non-trivial if only if sk > nk p, q p p, q p 2 , k = 1, ··· , r. When p = q, we write Ls = Ls and As = As which are respectively the usual weighted Lebesgue space and the usual weighted Bergman space. Moreover, when p = q = 2 the orthogonal 2 2 projector Ps from the Hilbert space Ls onto its closed subspace As is called weighted Bergman projector. It is well known that Ps is the 2 integral operator on Ls given by the formula PsF (z)= Bs(z,u + iv)F (u + iv)∆s n (v)dudv, − r ZTΩ where z − u + iv Bs(z,u + iv)= ds∆−s− n ( ) r i 2 is the reproducing kernel on As, called weighted Bergman kernel of x+iy T . Precisely, ∆ s n ( ) is the holomorphic determination of the Ω − − r i n (−s − )-power which reduces to the function ∆ s n (y) when x = 0. r − − r Our first result is an atomic decomposition theorem for functions in mixed norm weighted Bergman spaces on tube domains over symmetric cones. It generalizes the result of [2] for usual weighted Bergman spaces on tube domains over symmetric cones and the result of [18] for mixed ATOMIC DECOMPOSITION AND COMPLEX INTERPOLATION 3 norm weighted Bergman spaces on the upper half-plane (the case n = r = 1). s r nk Theorem A. Let be a vector of R such that sk > 2 , k =1, ··· , r. p,q Assume that Ps extends to a bounded operator on Ls . Then there is a sequence of points {zl,j = xl,j +iyj}l∈Z, j∈N in TΩ and a positive constant C such that the following assertions hold. (i) For every sequence {λl,j}l∈Z, j∈N such that q p p |λl,j| ∆s nq (yj) < ∞, + rp j l ! X X the series λl,j∆s nq (yj)Bs(z, zl,j) + rp l,j X p,q is convergent in As . Moreover, its sum F satisfies the inequality q p q p kF k p,q ≤ C |λl,j| ∆s nq (yj) As + rp j l ! X X p,q (ii) Every function F ∈ As may be written as F (z)= λl,j∆s nq (yj)Bs(z, zl,j), + rp l,j X with q p p q |λl,j| ∆s nq (yj) ≤ CkF k p,q + rp As j l ! X X Our second result is an interpolation theorem between mixed norm weighted Bergman spaces. It generalizes the result of [5] for usual weighted Bergman spaces. We adopt the following notation. nk sk − 2 qs = min 1+ m . 1≤k≤r k 2 s s r n Theorem B. (1) Let 0, 1 ∈ R be such that (s0)k, (s1)k > r −1, k = 1, ··· ,r and let 1 ≤ p0,p1 ≤ ∞, 1 ≤ q0, q1 < ∞ be such that 1 ≤ qi < qsi for every i =0, 1. Then for every θ ∈ (0, 1), we have p0,q0 p1,q1 p,q [As0 , As1 ]θ = As 4 DAVID BEKOLLE, JOCELYN GONESSA AND CYRILLE NANA 1 1−θ θ 1 1−θ θ s with equivalent norms, where p = p0 + p1 , q = q0 + q1 and q = (1−θ)s0 θs1 q0 + q1 . r n s R k s (2) Let ∈ be such that sk > 2 , k = 1, ··· , r. Assume that P pi,qi extends to a bounded operator on Ls , i =0, 1 for 1 ≤ p0,p1 ≤∞ and 1 < q0, q1 < ∞. Then for every θ ∈ (0, 1), we have p0,q0 p1,q1 p,q [As , As ]θ = As 1 1−θ θ 1 1−θ θ with equivalent norms, where p = p0 + p1 and q = q0 + q1 . s r n (3) Let ∈ R be such that (s)k > r − 1, k = 1, ··· , r, let 1 ≤ p0 < p1 ≤∞ and let q0, q1 be such that 1 ≤ q0 < qs ≤ q1. We assume that p1,q1 Ps extends to a bounded operator on Ls . Then for some values of θ ∈ (0, 1), we have p0,q0 p1,q1 p,q [As , As ]θ = As 1 1−θ θ 1 1−θ θ with equivalent norms, where p = p0 + p1 and q = q0 + q1 . We recall sufficient conditions on p, q and s under which the weighted p,q Bergman projector Ps extends to a bounded operator on Ls . We adopt the following notations. n sk + r ps = 1 + min k (mk − sk)+ and ′ qs(p) = min{p,p }qs. Theorem 1.1 ([12] and [17]). The weighted Bergman projector Ps ex- p,q 1 1 1 tends to a bounded operator on Ls whenever qs(p) < q < 1 − qs(p) in the following two cases: nj s (i) sj > 2 , j =1, ··· r and 1 ≤ p<p [12]; n (ii) sj > r − 1, j =1, ··· r and 1 ≤ p ≤∞ [17]. We restrict to tube domains over Lorentz cones (r = 2). For real s =(s,s), this problem was recently completely solved on tube domains over Lorentz cones (cf. [4] for a combination of results from [8] and [1]; n n cf. also [7] for the unweighted case s = ( r , ··· , r )). For vectorial s =(s1,s2), Theorem 1.1 was also extended in [4] to other values p and q. ATOMIC DECOMPOSITION AND COMPLEX INTERPOLATION 5 The plan of this paper is as follows. In Section 2, we overview some preliminaries and useful results about symmetric cones and tube do- mains over symmetric cones. In Section 3, we study atomic decom- position of mixed norm Bergman spaces and we prove a more precise statement of Theorem A. In Section 4, we study interpolation via the complex method between mixed norm weighted Bergman spaces and we prove Theorem B. In particular we give a more precise statement of assertion (3) of this theorem (Theorem 4.6) and we ask an open question. A final remark will point out a connection between the two main theorems of the paper (Theorem A and Theorem B). For s = (s, ··· ,s) real, Theorem A and Theorem B were presented in the PhD dissertation of the second author [14]. 2. Preliminaries Materials of this section are essentially from [13]. We give some definitions and useful results. Let Ω be an irreducible symmetric cone of rank r in a real vector space V of dimension n endowed with the structure of Euclidean Jordan algebra with identity e. In particular, Ω is self-dual with respect to the inner product (x|y)= tr(xy) on V . 2.1. Group action. Let G(Ω) be the group of linear transformations of the cone Ω and G its identity component. By definition, the subgroup G of G(Ω) is a semi-simple Lie group which acts transitively on Ω. This gives the identification Ω ∼ G/K, where K := {g ∈ G : g · e = e} is a maximal compact subgroup of G. More precisely, K = G ∩ O(V ), where O(V ) is the orthogonal group in V. Furthermore, there is a solv- able subgroup T of G acting simply transitively on Ω. That is, every y ∈ Ω can be written uniquely as y = t · e, for some t ∈ T.

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