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Weighted Bergman Spaces Induced by Rapidly Increasing Weights
José Ángel Peláez, Universidad de Málaga, Spain, and Jouni Rättyä, University of Eastern Finland, Joensuu, Finland
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Memoirs of the American Mathematical Society
2014; 124 pp; softcover
Volume: 227
ISBN-10: 0-8218-8802-1
ISBN-13: 978-0-8218-8802-5
List Price: US$77 Individual Members: US$46.20
Institutional Members: US\$61.60
Order Code: MEMO/227/1066

This monograph is devoted to the study of the weighted Bergman space $$A^p_\omega$$ of the unit disc $$\mathbb{D}$$ that is induced by a radial continuous weight $$\omega$$ satisfying $$\lim_{r\to 1^-}\frac{\int_r^1\omega(s)\,ds}{\omega(r)(1-r)}=\infty.$$ Every such $$A^p_\omega$$ lies between the Hardy space $$H^p$$ and every classical weighted Bergman space $$A^p_\alpha$$. Even if it is well known that $$H^p$$ is the limit of $$A^p_\alpha$$, as $$\alpha\to-1$$, in many respects, it is shown that $$A^p_\omega$$ lies "closer" to $$H^p$$ than any $$A^p_\alpha$$, and that several finer function-theoretic properties of $$A^p_\alpha$$ do not carry over to $$A^p_\omega$$.

• Preface
• Basic notation and introduction to weights
• Description of $$q$$-Carleson measures for $$A^p_\omega$$
• Factorization and zeros of functions in $$A^p_\omega$$
• Integral Operators and equivalent norms
• Non-conformally invariant space induced by $$T_g$$ on $$A^p_\omega$$
• Schatten classes of the integral operator $$T_g$$ on $$A^2_\omega$$
• Applications to differential equations
• Further discussion
• Bibliography
• Index