Boundedness of a family of Hilbert-type operators and its Bergman-type analogue

Abstract

In this paper, we first consider boundedness properties of a family of operators generalizing the Hilbert operator in the upper triangle case. In the diagonal case, we give the exact norm of these operators under some restrictions on the parameters. We secondly consider boundedness properties of a family of positive Bergman-type operators of the upper-half plane. We give necessary and sufficient conditions on the parameters under which these operators are bounded in the upper triangle case.2010 Mathematics Subject Classification. Primary 47B34, 26D15; Secondary 28A25.

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

“…Our main tools for the sufficient parts will be a off-diagonal Schur's test due to G. O. Okikiolu and integrability conditions of the determinant function among others. We also refer to [1,20] for the corresponding one dimension results and applications.…”

“…Let α, β, γ be real parameters. We consider the integral operators S = S α,β,γ which are defined on Ω by The above family can be seen as a generalization of the Hilbert-type operators considered in [1]. It has been considered in several works related to the question of boundedness of Bergman operators (see for example [3,4,19] and the references therein).…”

“…The aim of this paper is to extend to the setting of the tube domains over symmetric cones, the L p − L q estimates of the Bergman projector, when 1 ≤ p ≤ q ≤ ∞, considered by authors of [1], who studied the case of upper half plane. This leads us to more general result than those of [19].…”