Mean Oscillations and Equimeasurable Rearrangements of Functions

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.

• Gurov – Reshetnyak classes
• Maximum
• bounded mean oscillation
• bounded mean oscillation (BMO)
• equimeasurable rearrangements of functions
• mean oscillation
• reverse Hölder inequalities

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)

• Institute of Mathematics, Economics and Mechanics, Odessa National I.I. Mechnikov University, Odessa, Ukraine

  Anatolii Korenovskii

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