Weighted inequalities of Hardy type. 2nd updated edition. (English) Zbl 1380.26001

Kufner, Alois; Persson, Lars-Erik; Samko, Natasha

Hackensack, NJ: World Scientific (ISBN 978-981-3140-64-6/hbk). xx, 459 p. (2017).

This book is devoted to the study of weighted inequalities of Hardy type in weighted Lebesgue spaces

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. Specifically, the characterization of weights for which the Hardy type operators are bounded in weighted Lebesgue are studied and discussed. In addition, new developments which complement and extend the previous results in the first edition of this book are included. In particular, the powerful convexity approach to prove Hardy-type inequalities which obviously was not discovered by G. H. Hardy in those dramatic years were presented, proved and well discussed in this book.
This 459 page book consists of seven chapters. Chapter 1 gives a brief overview of weighted Lebesgue spaces and a survey of the various characterizations of Hardy-type inequalities in weighted Lebesgue spaces. The chapter also includes some results on Hardy-type inequalities with derivatives and some results concerning the compactness property of the Hardy operator. In Chapter 2, some natural extensions of Hardy operator usually called the Hardy-type operators are investigated. Special cases of these operators such as the Riemann-Liouville fractional integral operator of order greater than or equal to one and its conjugate, Weyl fractional operators are examined and the weight characterizations for which these operators are bounded on weighted Lebesgue spaces are deduced from these general results. Chapter 3 deals with Hardy-Steklov operator, its various weight characterizations and applications to the study of financial markets are given. Chapter 4 is devoted to weighted Hardy inequalities

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in differential form

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even with higher-order derivatives while Chapter 5 deals with fractional order Hardy inequalities in a weighted setting. Chapter 6 examines integral operators on the cone of monotone functions

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and obtains its weight characterizations. Finally, the last chapter of this book which is Chapter 7 presents some results concerning Hardy-type inequalities in function spaces other than the weighted Lebesgues spaces. Furthermore, the powerful convexity approach to prove Hardy-type inequalities was presented and discussed. Also, some new multidimensional results on Hardy type inequalities are included with some open questions for further study.
The book is well written and the authors present historical facts and new results on Hardy-type inequalities. In addition, each chapter of this book ends with comments and remarks which include open questions and references for the reader. This book is recommended for graduate students, researchers and analysts hunting for inequalities to use and cannot prove.

Reviewer: James Adedayo Oguntuase (Abeokuta)

MSC:

26-02	Research exposition (monographs, survey articles) pertaining to real functions
26D10	Inequalities involving derivatives and differential and integral operators
47G10	Integral operators

Keywords:

Hardy operator; weighted inequalities convexity approach

Citations:

Zbl 1065.26018