Mean Oscillations and Equimeasurable Rearrangements of Functions [Paperback]

This volume considers various applications of equimeasurable function rearrangements to the best constant -type problems. It presents several classical theorems along with some very recent results. Coverage includes a product-space extension of the Rising Sun lemma, a product-space version of the John-Nirenberg inequality for bounded mean oscillation functions with sharp exponent, and sharp embedding theorems for Muckenhoupt, Gurov-Reshetnyak, and Gehring classes.

From the reviews:

This book is devoted to classes (spaces) of functions that can be described in terms of mean oscillations. & The sharp constants in the corresponding relations have been found in a number of works by the author of the book under review and his students; these works form the core of the present book. & The book is well written. We mention that it contains many examples with full and very careful calculations. & a good and convenient source for anyone interested in the area. (Andrei K. Lerner, Mathematical Reviews, Issue 2008 k)

Various applications of equimeasurable function rearrangements to the ''best constant -type problems are considered in this volume. Several classical theorems are presented along with some very recent results. In particular, the text includes a product-space extension of the Rising Sun lemma, a product-space version of the John-Nirenberg inequality for  bounded mean oscillation (BMO) functions with sharp exponent, a refinement of the Gurov-Reshetnyak lemma, sharp embedding theorems for Muckenhoupt, Gurov-Reshetnyak, reverse H?lder, and Gehring classes, etc. This volume is interesting for graduate students and mathematicians il3l