Cyrille Nana

Some questions related to the Bergman projection in symmetric domains

Abstract This is essentially a survey on tube domains over irreducible symmetric cones or their bounded realizations. It includes the fact that in rank 2 these inequalities have now been proved for the whole range of possible exponents p, due to recent work of Bourgain and Demeter. We also discuss Lp estimates with loss for the bounded domains and conclude with some remarks and questions related to Bloch functions.

Korányi's lemma for homogeneous Siegel domains of type II. Applications and extended results

We show that the modulus of the Bergman kernel B(z, ζ) of a general homogeneous Siegel domain of type II is ”almost constant” uniformly with respect to z when ζ varies inside a Bergman ball. The control is expressed in terms of the Bergman distance. This result was proved by A. Koranyi for symmetric Siegel domains of type II. Subsequently, R. R. Coifman and R. Rochberg used this result to establish an atomic decomposition theorem and an interpolation theorem by functions in Bergman spaces \(A^p\) on these domains. The atomic decomposition theorem and the interpolation theorem are extended here to the general homogeneous case using the same tools. We further extend the range of exponents \(p\) via functional analysis using recent estimates. DOI: 10.1017/S0004972714000033

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

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.

Off-diagonal estimates of some Bergman-type operators of tube domains over symmetric cones

We obtain some necessary and sufficient conditions for the boundedness of a family of positive operators defined on symmetric cones, we then deduce off-diagonal boundedness of associated Bergman-type operators in tube domains over symmetric cones.