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Hardy Inequalities on Homogeneous Groups

100 Years of Hardy Inequalities

  • Book
  • Open Access
  • © 2019

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  • Presents a step-by-step guide for the techniques of basic functional inequalities from the point of view of Folland and Stein's homogeneous (Lie) groups, and for the applications of such methods. In addition, this book shows that these methods sometimes give new results even in classical (Euclidean) cases
  • Aims to collect the ideas underpinning Hardy type inequalities on general homogeneous groups, in a way, accessible to anyone with a basic level of understanding of analysis
  • Provides a self-contained coverage of elements of the traditional and modern analysis on homogeneous Lie groups, and does not require a previous background in Lie theory
  • Represents a detailed account of the recent developments in the field of anisotropic functional inequalities and their links to potential and other properties of operators

Part of the book series: Progress in Mathematics (PM, volume 327)

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About this book

This open access book provides an extensive treatment of Hardy inequalities and closely related topics from the point of view of Folland and Stein's homogeneous (Lie) groups. The place where Hardy inequalities and homogeneous groups meet is a beautiful area of mathematics with links to many other subjects. While describing the general theory of Hardy, Rellich, Caffarelli-Kohn-Nirenberg, Sobolev, and other inequalities in the setting of general homogeneous groups, the authors pay particular attention to the special class of stratified groups. In this environment, the theory of Hardy inequalities becomes intricately intertwined with the properties of sub-Laplacians and subelliptic partial differential equations. These topics constitute the core of this book and they are complemented by additional, closely related topics such as uncertainty principles, function spaces on homogeneous groups, the potential theory for stratified groups, and the potential theory for general Hörmander's sums of squares and their fundamental solutions.

This monograph is the winner of the 2018 Ferran Sunyer i Balaguer Prize, a prestigious award for books of expository nature presenting the latest developments in an active area of research in mathematics. As can be attested as the winner of such an award, it is a vital contribution to literature of analysis not only because it presents a detailed account of the recent developments in the field, but also because the book is accessible to anyone with a basic level of understanding of analysis. Undergraduate and graduate students as well as researchers from any field of mathematical and physical sciences related to analysis involving functional inequalities or analysis of homogeneous groups will find the text beneficial to deepen their understanding.

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Table of contents (13 chapters)


“This book is devoted to Hardy inequalities and similar inequalities, Rellich, Sobolev, Caffarelli-Kohn-Nirenberg inequalities on homogeneous Lie groups. … The book is a well written exhaustive monograph of the subject. It contains also a rich bibliography.” (Leszek Skrzypczak, zbMATH 1428.22011, 2020)

Authors and Affiliations

  • Department of Mathematics, Imperial College London, UK; Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium; School of Mathematical Sciences, Queen Mary University of London, London, UK

    Michael Ruzhansky

  • Department of Mathematics, Nazarbayev University, Astana, Kazakhstan

    Durvudkhan Suragan

About the authors

Michael Ruzhansky is a Professor of Pure Mathematics at Imperial College London.

Durvudkhan Suragan is an Assistant Professor of Mathematics at Nazarbayev University.

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