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Softcover ISBN:  9781470442132 
Product Code:  MEMO/266/1293 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
eBook ISBN:  9781470462499 
Product Code:  MEMO/266/1293.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Softcover ISBN:  9781470442132 
eBook ISBN:  9781470462499 
Product Code:  MEMO/266/1293.B 
List Price:  $170.00 $127.50 
MAA Member Price:  $153.00 $114.75 
AMS Member Price:  $136.00 $102.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 266; 2020; 97 ppMSC: Primary 42; 31
Fix \(d\geq 2\), and \(s\in (d1,d)\). The authors characterize the nonnegative locally finite nonatomic Borel measures \(\mu \) in \(\mathbb R^d\) for which the associated \(s\)Riesz transform is bounded in \(L^2(\mu )\) in terms of the Wolff energy. This extends the range of \(s\) in which the MateuPratVerdera characterization of measures with bounded \(s\)Riesz transform is known.
As an application, the authors give a metric characterization of the removable sets for locally Lipschitz continuous solutions of the fractional Laplacian operator \((\Delta )^\alpha /2\), \(\alpha \in (1,2)\), in terms of a wellknown capacity from nonlinear potential theory. This result contrasts sharply with removability results for Lipschitz harmonic functions.

Table of Contents

Chapters

1. Introduction

2. Preliminaries

3. The general scheme: Finding a large Lipschitz oscillation coefficient

4. Upward and Downward Domination

Part I: The blowup procedures

5. Preliminary results regarding reflectionless measures

6. The basic energy estimates

7. Blow up I: The density drop

8. The choice of the shell

9. Blow up II: Doing away with $\varepsilon $

10. Localization around the shell

Part II: The nonexistence of an impossible object

11. The scheme

12. Suppressed kernels

13. Step I: CalderónZygmund theory (From a distribution to an $L^p$function)

14. Step II: The smoothing operation

15. Step III: The variational argument

16. Contradiction

Appendices

A. The maximum principle

B. The small boundary mesh

C. Lipschitz continuous solutions of the fractional Laplacian equation

D. Index of Selected Symbols and Terms


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Fix \(d\geq 2\), and \(s\in (d1,d)\). The authors characterize the nonnegative locally finite nonatomic Borel measures \(\mu \) in \(\mathbb R^d\) for which the associated \(s\)Riesz transform is bounded in \(L^2(\mu )\) in terms of the Wolff energy. This extends the range of \(s\) in which the MateuPratVerdera characterization of measures with bounded \(s\)Riesz transform is known.
As an application, the authors give a metric characterization of the removable sets for locally Lipschitz continuous solutions of the fractional Laplacian operator \((\Delta )^\alpha /2\), \(\alpha \in (1,2)\), in terms of a wellknown capacity from nonlinear potential theory. This result contrasts sharply with removability results for Lipschitz harmonic functions.

Chapters

1. Introduction

2. Preliminaries

3. The general scheme: Finding a large Lipschitz oscillation coefficient

4. Upward and Downward Domination

Part I: The blowup procedures

5. Preliminary results regarding reflectionless measures

6. The basic energy estimates

7. Blow up I: The density drop

8. The choice of the shell

9. Blow up II: Doing away with $\varepsilon $

10. Localization around the shell

Part II: The nonexistence of an impossible object

11. The scheme

12. Suppressed kernels

13. Step I: CalderónZygmund theory (From a distribution to an $L^p$function)

14. Step II: The smoothing operation

15. Step III: The variational argument

16. Contradiction

Appendices

A. The maximum principle

B. The small boundary mesh

C. Lipschitz continuous solutions of the fractional Laplacian equation

D. Index of Selected Symbols and Terms