Morrey spaces
type: Item
instance of: scholarly article
title: Morrey spaces
zbMATH Open document ID: 1339.42001
DOI: 10.1007/978-3-319-26681-7
author: David R. Adams
published in: Applied and Numerical Harmonic Analysis
publication date: 2015-12-09T00:00:00Z
review text: In this book, the author provides some of the latest research on the theory of Morrey spaces associated with contemporary harmonic analysis. Let \(p\in(1,\infty)\) and \(\lambda\in[0,n]\). Recall that a function \(f\) on \(\mathbb{R}^n\) belongs to the Morrey space \(L^{p,\lambda}(\mathbb{R}^n)\) if \[ \sup_{x\in\mathbb{R}^n,\,r\in(0,\infty)}\left[r^{\lambda-n}\int_{B(x,r)}|f(y)|^p\,dy\right]^{1/p}<\infty. \] In Chapter 1 of this book, the author gives a brief introduction to the history of Morrey spaces and the main contents included in this book. Then, in Chapter 2, several classical function spaces which have a close relationship with Morrey spaces are presented. To state the (pre-)dual theory of Morrey spaces in Chapter 5, the author naturally reviews the notions of the Hausdorff capacity and the Choquet integral with respect to the Hausdorff capacity in Chapters 3 and 4, respectively. Moreover, the boundedness of some classical operators in harmonic analysis, including the Hardy-Littlewood maximal operator, potential operators and Calderón-Zygmund singular integral operators on Morrey spaces is given in Chapters 6, 7 and 8. In Chapter 11, the interpolation result for Morrey spaces is presented. In further chapters, the author gives some applications of the theory of Morrey spaces to: Morrey-Sobolev capacities (Chapter 9), traces of Morrey potentials with the multiplier operator (Chapter 10), commutators of Morrey potentials (Chapter 12), Mock Morrey spaces (Chapter 13), Morrey-Besov spaces and Besov capacity (Chapter 14) and non-linear elliptic PDE (Chapters 15 and 16). Finally, in Chapter 17 of this book, Morrey spaces on complete Riemannian manifolds are discussed. This book contains the latest results obtained by the author. It is a useful reference to mathematicians working in potential theory, harmonic analysis and partial differential equations, in particular, in Morrey spaces theory.
Mathematics Subject Classification ID: 42-02
Mathematics Subject Classification ID: 42B35
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42B37
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 46B70
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 6518661
zbMATH Keywords: Morrey spaces
zbMATH Keywords: Hardy-Littlewood maximal operator
zbMATH Keywords: potential operators
zbMATH Keywords: Calderón-Zygmund singular integral operators
zbMATH Keywords: interpolation
zbMATH Keywords: Morrey-Sobolev capacities
zbMATH Keywords: Morrey-Besov spaces
zbMATH Keywords: Riemannian manifolds
zbMATH Keywords: non-linear elliptic PDEs
reviewed by: Sibei Yang
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-26681-7
OpenAlex ID: W4230410641
description: scientific article; zbMATH DE number 6518661