L ^p-L ^q Estimates for Bergman Projections in Bounded Symmetric Domains of Tube Type

Abstract

Let D be an irreducible bounded symmetric domain of tube type in ℂⁿ. The class of Bloch functions is well known in this context, in connection with Hankel operators or duality of Bergman spaces. Contrary to what happens in the unit ball, Bloch functions do not belong to all Lebesgue spaces L ^p(D) for p<∞ in higher rank. We give here both necessary and sufficient conditions on p for such an embedding. This question is equivalent to local boundedness properties of the Bergman projection in the tube domain over a symmetric cone that is conformally equivalent to D. We are linked to consider L ^∞–L ^q inequalities on symmetric cones, which may be of independent interest, and study more systematically estimates with loss for the Bergman projection. The proofs are based on a very precise estimate on an integral related to the Gamma function of a symmetric cone.

Appendix

1.1 A.1 A Remark on Quaternionic Determinants

As pointed out in the proof of Lemma 4.10, the determinant of a matrix of quaternions (for which actually there are various definitions) is in general not multilinear; see, e.g., [2, 9]. Thus, we briefly explain here what definition must be used in order to justify (4.15) in the case V=Her(r,ℍ).

A matrix of quaternions $A=(a_{ij})_{i,j=1}^{n}\in M_{n\times n}(\mathbb {H})$ is called almost-Hermitian if there exists (at most) one index k∈{1,…,n} so that

$$a_{ij}=\bar{a}_{ji},\quad\forall i,j\in\{1,\ldots,n\} \setminus\{k\}. $$

For such matrices the Moore determinant is defined by

$$\mbox{Mdet}(A) = \sum_{\ell=1}^n \varepsilon _{k\ell}a_{k\ell}\mbox{Mdet}\bigl(A[k,\ell]\bigr), $$

where k is the index in the definition of almost-Hermitian, A[k,ℓ] is the matrix obtained from A by first interchanging the ℓ-th and k-th columns, and then deleting both the k-th row and k-th column, and ε _kℓ =−1 if $\ell\not=k$ and ε _kk =1. We refer to [9, 14] for the consistency of this definition and various properties of such determinants. It is easy to verify from the definition that, if A,B,C are almost-Hermitian matrices with the same index k, and satisfying the linear relation

$$c_{kj}=a_{kj}+b_{kj}\quad\forall j,{\quad \mbox {and}\quad }c_{ij}=a_{ij}=b_{ij}\quad\forall i\not=k,\ \forall j, $$

then

$$\mbox{Mdet}(C)=\mbox{Mdet}(A)+\mbox{Mdet}(B) $$

(see, e.g., [9, Theorem 2]). Using this property and the definition of Mdet, it is straightforward to justify the analogue of (4.15) for quaternionic matrices, i.e.,

$$\mbox{Mdet} \bigl(\mathbf {s}+\bigl(\mathbf {v}^2\bigr)' \bigr) = s_1\cdots s_{r-1}+\sum_{j=1}^{r-1} s_1\cdots|v_j|^2\cdots s_{r-1}, $$

when s _j >0 and v _j ∈ℍ.

Finally, we remark that the Jordan algebra determinant Δ(x) of x∈Her(r,ℍ) (defined as the independent coefficient of the minimal polynomial of x; see [10, Chap. II]) coincides with the Moore determinant Mdet(x) defined above. This follows, for instance, from the formulation of each of these determinants as Pfaffians of matrices in Skew(2r,ℂ) (see [10, p. 40] and [9, (4.7)]).