Softcover ISBN:  9781470465759 
Product Code:  SURV/257 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
eBook ISBN:  9781470466602 
Product Code:  SURV/257.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Softcover ISBN:  9781470465759 
eBook: ISBN:  9781470466602 
Product Code:  SURV/257.B 
List Price:  $250.00 $187.50 
MAA Member Price:  $225.00 $168.75 
AMS Member Price:  $200.00 $150.00 
Softcover ISBN:  9781470465759 
Product Code:  SURV/257 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
eBook ISBN:  9781470466602 
Product Code:  SURV/257.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Softcover ISBN:  9781470465759 
eBook ISBN:  9781470466602 
Product Code:  SURV/257.B 
List Price:  $250.00 $187.50 
MAA Member Price:  $225.00 $168.75 
AMS Member Price:  $200.00 $150.00 

Book DetailsMathematical Surveys and MonographsVolume: 257; 2021; 354 ppMSC: Primary 42; 46; Secondary 26; 28; 31; 35
This book discusses advances in maximal function methods related to Poincaré and Sobolev inequalities, pointwise estimates and approximation for Sobolev functions, Hardy's inequalities, and partial differential equations. Capacities are needed for fine properties of Sobolev functions and characterization of Sobolev spaces with zero boundary values. The authors consider several uniform quantitative conditions that are selfimproving, such as Hardy's inequalities, capacity density conditions, and reverse Hölder inequalities. They also study Muckenhoupt weight properties of distance functions and combine these with weighted norm inequalities; notions of dimension are then used to characterize density conditions and to give sufficient and necessary conditions for Hardy's inequalities. At the end of the book, the theory of weak solutions to the \(p\)Laplace equation and the use of maximal function techniques is this context are discussed.
The book is directed to researchers and graduate students interested in applications of geometric and harmonic analysis in Sobolev spaces and partial differential equations.
ReadershipGraduate students and researchers interested in functional analysis.

Table of Contents

Chapters

Maximal functions

Lipschitz and Sobolev functions

Sobolev and Poincaré inequalities

Pointwise inequalities for Sobolev functions

Capacities and fine properties of Sobolev functions

Hardy’s inequalities

Density conditions

Muckenhoupt weights

Weighted maximal and Poincaré inequalities

Distance weights and Hardy–Sobolev inequalities

The $p$Laplace equation

Stability results for the $p$Laplace equation


Additional Material

Reviews

The book is written in a concise style, but is detailed enough for a good understanding of techniques and proofs. Most of the material is selfcontained, only requiring elements of functional analysis and measure and integration theory.
Alexandre Almeida (University of Aveiro), MathSciNet Reviews


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This book discusses advances in maximal function methods related to Poincaré and Sobolev inequalities, pointwise estimates and approximation for Sobolev functions, Hardy's inequalities, and partial differential equations. Capacities are needed for fine properties of Sobolev functions and characterization of Sobolev spaces with zero boundary values. The authors consider several uniform quantitative conditions that are selfimproving, such as Hardy's inequalities, capacity density conditions, and reverse Hölder inequalities. They also study Muckenhoupt weight properties of distance functions and combine these with weighted norm inequalities; notions of dimension are then used to characterize density conditions and to give sufficient and necessary conditions for Hardy's inequalities. At the end of the book, the theory of weak solutions to the \(p\)Laplace equation and the use of maximal function techniques is this context are discussed.
The book is directed to researchers and graduate students interested in applications of geometric and harmonic analysis in Sobolev spaces and partial differential equations.
Graduate students and researchers interested in functional analysis.

Chapters

Maximal functions

Lipschitz and Sobolev functions

Sobolev and Poincaré inequalities

Pointwise inequalities for Sobolev functions

Capacities and fine properties of Sobolev functions

Hardy’s inequalities

Density conditions

Muckenhoupt weights

Weighted maximal and Poincaré inequalities

Distance weights and Hardy–Sobolev inequalities

The $p$Laplace equation

Stability results for the $p$Laplace equation

The book is written in a concise style, but is detailed enough for a good understanding of techniques and proofs. Most of the material is selfcontained, only requiring elements of functional analysis and measure and integration theory.
Alexandre Almeida (University of Aveiro), MathSciNet Reviews