Lecture Notes in Mathematics

4 primary works

Morrey and Campanato Meet Besov, Lizorkin and Triebel

by Wen Yuan, Winfried Sickel, and Dachun Yang

Published 4 September 2010

During the last 60 years the theory of function spaces has been a subject of growing interest and increasing diversity. Based on three formally different developments, namely, the theory of Besov and Triebel-Lizorkin spaces, the theory of Morrey and Campanato spaces and the theory of Q spaces, the authors develop a unified framework for all of these spaces. As a byproduct, the authors provide a completion of the theory of Triebel-Lizorkin spaces when p = ∞.

The Hardy Space H1 with Non-Doubling Measures and Their Applications

by Dachun Yang, Dongyong Yang, and Guoen Hu

Published 13 January 2014

The present book offers an essential but accessible introduction to the discoveries first made in the 1990s that the doubling condition is superfluous for most results for function spaces and the boundedness of operators. It shows the methods behind these discoveries, their consequences and some of their applications. It also provides detailed and comprehensive arguments, many typical and easy-to-follow examples, and interesting unsolved problems.

The theory of the Hardy space is a fundamental tool for Fourier analysis, with applications for and connections to complex analysis, partial differential equations, functional analysis and geometrical analysis. It also extends to settings where the doubling condition of the underlying measures may fail.

Real-Variable Theory of Musielak-Orlicz Hardy Spaces

by Dachun Yang, Yiyu Liang, and Luong Dang Ky

Published 10 May 2017

The main purpose of this book is to give a detailed and complete survey of recent progress related to the real-variable theory of Musielak-Orlicz Hardy-type function spaces, and to lay the foundations for further applications.

The real-variable theory of function spaces has always been at the core of harmonic analysis. Recently, motivated by certain questions in analysis, some more general Musielak-Orlicz Hardy-type function spaces were introduced. These spaces are defined via growth functions which may vary in both the spatial variable and the growth variable. By selecting special growth functions, the resulting spaces may have subtler and finer structures, which are necessary in order to solve various endpoint or sharp problems.

This book is written for graduate students and researchers interested in function spaces and, in particular, Hardy-type spaces.

Gaussian Capacity Analysis

by Liguang Liu, Jie Xiao, Dachun Yang, and Wen Yuan

Published 21 September 2018

This monograph develops the Gaussian functional capacity theory with applications to restricting the Gaussian Campanato/Sobolev/BV space. Included in the text is a new geometric characterization of the Gaussian 1-capacity and the Gaussian Poincare 1-inequality. Applications to function spaces and geometric measures are also presented.

This book will be of use to researchers who specialize in potential theory, elliptic differential equations, functional analysis, probability, and geometric measure theory.