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Mean Oscillations and Equimeasurable Rearrangements of Functions : Lecture Notes Of The Unione Matematica Italiana - Anatolii Korenovskii

Mean Oscillations and Equimeasurable Rearrangements of Functions

Lecture Notes Of The Unione Matematica Italiana

Paperback Published: 19th September 2007
ISBN: 9783540747086
Number Of Pages: 188

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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)

Preliminaries and Auxiliary Resultsp. 1
Equimeasurable Rearrangements of Functionsp. 1
Covering Lemmasp. 3
Properties of Oscillations and the Definition of the BMO-classp. 25
Properties of Mean Oscillationsp. 25
Definition of the BMO-class and Examplesp. 39
Estimates of Rearrangements and the John-Nirenberg Theoremp. 61
Estimates of Rearrangements of the BMO-functionsp. 61
The John-Nirenberg Inequalityp. 72
One-Dimensional Casep. 77
Anisotropic Casep. 79
The BMO-estimates of the Hardy-type Transformsp. 81
Estimates of Oscillations of the Hardy Transformp. 81
Estimates of the Oscillations of the Conjugate Hardy Transform and the Calderon Transformp. 92
The Gurov-Reshetnyak Class of Functionsp. 99
Embedding in the Gehring Classp. 99
One-Dimensional Casep. 122
Anisotropic Casep. 131
Embedding in the Muckenhoupt Classp. 135
One-Dimensional Casep. 141
Anisotropic Casep. 145
The Boundedness of the Hardy-Littlewood Maximal Operator from BMO into BLOp. 147
The Weighted Analogs of the Riesz Lemma and the Gurov-Reshetnyak Theoremp. 151
The Weighted Riesz Lemmap. 151
The Gurov-Reshetnyak Theorem in the Weighted Casep. 152
Classes of Functions Satisfying the Reverse Holder Inequalityp. 157
Estimate of Rearrangements of Functions Satisfying the Reverse Jensen Inequalityp. 157
About the Exact Extension of the Reverse Weighted Holder Inequalityp. 171
Hardy's Inequalitiesp. 174
The Proof of the Main Theoremp. 177
Referencesp. 183
Indexp. 187
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9783540747086
ISBN-10: 3540747087
Series: Lecture Notes Of The Unione Matematica Italiana
Audience: General
Format: Paperback
Language: English
Number Of Pages: 188
Published: 19th September 2007
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.55 x 16.23  x 1.19
Weight (kg): 0.31