Mean Oscillations and Equimeasurable Rearrangements of Functions

ISBN-13: 9783540747086 / Angielski / Miękka / 2007 / 189 str.

Anatolii A. Korenovskii

Opis

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.

Recenzje

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)

Spis treści

Preliminaries and Auxiliary Results.- Properties of Oscillations and the Definition of the BMO-class.- Estimates of Rearrangements and the John–Nirenberg Theorem.- The BMO-estimates of the Hardy-type Transforms.- The Gurov–Reshetnyak Class of Functions.

Blurb

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 Hölder, and Gehring classes, etc. This volume is interesting for graduate students and mathematicians involved with these topics.