Morrey spaces
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.
Calderón-Zygmund singular integral operators
scientific article; zbMATH DE number 6518661