Carleson Embeddings and Two Operators on Bergman Spaces of Tube Domains over Symmetric Cones

Abstract

We prove Carleson embeddings for Bergman spaces of tube domains over symmetric cones, we apply them to characterize symbols of bounded Cesàro-type operators from weighted Bergman spaces to weighted Besov spaces. We also obtain Schatten class criteria of Toeplitz operators and Cesàro-type operators on weighted Hilbert-Bergman spaces.

Schatten class Toeplitz operators on weighted Bergman spaces of tube domains over symmetric cones

“…For unbounded domains, Schatten classes have been also characterized in Fock spaces by several authors (see for example [11,14] and the references therein). In [13], we extended these results for 1 ≤ p ≤ ∞ to weighted Bergman spaces of tube domains over symmetric cones. To be more precise, let us introduce more notations.…”

“…5. In the last section, we apply our results to extend to the range 1 ≤ p < 2 , the characterization of Schatten class S p (A 2 ν (D)) for the Cesàro-type operator obtained in [13].…”

“…For z ∈ D and δ ∈ (0, 1) , we define the average of the positive measure µ at z by Consider the measure d (z) = � −2n/r (Iz) dV (z). The following was obtained in [13]. Theorem 1.1 Let µ be a positive Borel measure on D, and ν > n r − 1.…”

“…We are essentially motivated here by the idea of extending the results in [13] for Toeplitz operators T µ in Schatten classes S p (A 2 ν (D)) for p ≥ 1 , to the case 0 < p < 1 , and settling the problem of the characterization of Schatten class S p (A 2 ν (D)) for 1 ≤ p < 2 for the Cesàro-type operator introduced in [13].…”

“…Recall that δ > 0, is the Bergman ball centered at z with radius δ , where d(·, ·) is the Bergman distance (for a definition, see for example [13]). We recall the following (see [2,Theorem 5.4]).…”