A class of Berezin-type operators on weighted Fock spaces with A_∞-type weights

2024/09/02 by Jiale Chen, Chen, Jiale
Mathematics · #Holomorphic and Operator Theory #Advanced Harmonic Analysis Research #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.2409.01132

Abstract

Let 0<α,β,t<∞ and μ be a positive Borel measure on ℂⁿ. We consider the Berezin-type operator S^t,α,β_μ defined by S^t,α,β_μf(z):=(∫_ℂⁿe^-\fracβ2|z-u|²|f(u)|^te^{-(αt)/(2)|u|2}dμ(u))^1/t, z∈ℂⁿ. We completely characterize the boundedness and compactness of S^t,α,β_μ from the weighted Fock space F^p_α,w into the Lebesgue space L^q(wdv) for all possible indices, where w is a weight on ℂⁿ that satisfies an A_∞-type condition. This solves an open problem raised by Zhou, Zhao and Tang [Banach J. Math. Anal. 18 (2024), Paper No. 20]. As an application, we obtain the description of the boundedness and compactness of Toeplitz-type operators acting between weighted Fock spaces induced by A_∞-type weights.

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