# Bergman spaces with exponential type weights

Ontology type: schema:ScholarlyArticle      Open Access: True

### Article Info

DATE

2021-12-11

AUTHORS ABSTRACT

For 1≤p<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1\le p<\infty$\end{document}, let Aωp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A^{p}_{\omega }$\end{document} be the weighted Bergman space associated with an exponential type weight ω satisfying ∫D|Kz(ξ)|ω(ξ)1/2dA(ξ)≤Cω(z)−1/2,z∈D,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \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 Kz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$K_{z}$\end{document} is the reproducing kernel of Aω2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A^{2}_{\omega }$\end{document}. 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 ω∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\omega _{*}$\end{document}. As an application, we prove the boundedness of the Bergman projection on Lωp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{p}_{\omega }$\end{document}, identify the dual space of Aωp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A^{p}_{\omega }$\end{document}, and establish an atomic decomposition for it. Further, we give necessary and sufficient conditions for the boundedness and compactness of some operators acting from Aωp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A^{p}_{\omega }$\end{document} into Aωq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A^{q}_{\omega }$\end{document}, 1≤p,q<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1\le p,q<\infty$\end{document}, such as Toeplitz and (big) Hankel operators. More... »

PAGES

193

### References to SciGraph publications

• 1992-05. Hankel and Toeplitz operators on Dirichlet spaces in INTEGRAL EQUATIONS AND OPERATOR THEORY
• 1998-12. Beurling-type density theorems for weightedLp spaces of entire functions in JOURNAL D'ANALYSE MATHÉMATIQUE
• 1993-10. Boundedness, compactness, and Schattenp-classes of Hankel operators between weighted Dirichlet spaces in ARKIV FÖR MATEMATIK
• 1993-12. Beurling type density theorems in the unit disk in INVENTIONES MATHEMATICAE
• 2013-10-11. Hankel Operators on Fock Spaces in CONCRETE OPERATORS, SPECTRAL THEORY, OPERATORS IN HARMONIC ANALYSIS AND APPROXIMATION
• 1995-12. Hankel operators on the weighted Bergman spaces with exponential type weights in INTEGRAL EQUATIONS AND OPERATOR THEORY
• 1978-02. Embedding theorems for weighted classes of harmonic and analytic functions in JOURNAL OF MATHEMATICAL SCIENCES
• 1988-12. Hankel operators between weighted Bergman spaces in ARKIV FÖR MATEMATIK
• 2003. Hankel Operators and Their Applications in NONE
• ### Journal

TITLE

Journal of Inequalities and Applications

ISSUE

1

VOLUME

2021

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1186/s13660-021-02726-4

DOI

http://dx.doi.org/10.1186/s13660-021-02726-4

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1143817449

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