Bergman spaces with exponential type weights

Abstract

For 1le p<infty , let A^{p}_{omega } be the weighted Bergman space associated with an exponential type weight ω satisfying \t\t\t∫D|Kz(ξ)|ω(ξ)1/2dA(ξ)≤Cω(z)−1/2,z∈D,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\int _{{\\mathbb{D}}} \\bigl\\vert K_{z}(\\xi ) \\bigr\\vert \\omega (\\xi )^{1/2} \\,dA(\\xi ) \\le C \\omega (z)^{-1/2}, \\quad z\\in {\\mathbb{D}}, $$\\end{document} where K_{z} is the reproducing kernel of A^{2}_{omega }. This condition allows us to obtain some interesting reproducing kernel estimates and more estimates on the solutions of the ∂̅-equation (Theorem 2.5) for more general weight omega _{*}. As an application, we prove the boundedness of the Bergman projection on L^{p}_{omega }, identify the dual space of A^{p}_{omega }, and establish an atomic decomposition for it. Further, we give necessary and sufficient conditions for the boundedness and compactness of some operators acting from A^{p}_{omega } into A^{q}_{omega }, 1le p,q<infty , such as Toeplitz and (big) Hankel operators.