eBook ISBN:  9781470414276 
Product Code:  MEMO/227/1066.E 
List Price:  $77.00 
MAA Member Price:  $69.30 
AMS Member Price:  $46.20 
eBook ISBN:  9781470414276 
Product Code:  MEMO/227/1066.E 
List Price:  $77.00 
MAA Member Price:  $69.30 
AMS Member Price:  $46.20 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 227; 2014; 124 ppMSC: Primary 30; Secondary 47
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 \(\lim_{r\to 1^}\frac{\int_r^1\omega(s)\,ds}{\omega(r)(1r)}=\infty.\) 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 \(\alpha\to1\), in many respects, it is shown that \(A^p_\omega\) lies “closer” to \(H^p\) than any \(A^p_\alpha\), and that several finer functiontheoretic properties of \(A^p_\alpha\) do not carry over to \(A^p_\omega\).

Table of Contents

Chapters

Preface

1. Basic Notation and Introduction to Weights

2. Description of $q$Carleson Measures for $A^p_\omega $

3. Factorization and Zeros of Functions in $A^p_\omega $

4. Integral Operators and Equivalent Norms

5. Nonconformally Invariant Space Induced by $T_g$ on $A^p_\omega $

6. Schatten Classes of the Integral Operator $T_g$ on $A^2_\omega $

7. Applications to Differential Equations

8. Further Discussion


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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 \(\lim_{r\to 1^}\frac{\int_r^1\omega(s)\,ds}{\omega(r)(1r)}=\infty.\) 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 \(\alpha\to1\), in many respects, it is shown that \(A^p_\omega\) lies “closer” to \(H^p\) than any \(A^p_\alpha\), and that several finer functiontheoretic properties of \(A^p_\alpha\) do not carry over to \(A^p_\omega\).

Chapters

Preface

1. Basic Notation and Introduction to Weights

2. Description of $q$Carleson Measures for $A^p_\omega $

3. Factorization and Zeros of Functions in $A^p_\omega $

4. Integral Operators and Equivalent Norms

5. Nonconformally Invariant Space Induced by $T_g$ on $A^p_\omega $

6. Schatten Classes of the Integral Operator $T_g$ on $A^2_\omega $

7. Applications to Differential Equations

8. Further Discussion