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This volume presents advances that have been made over recent decades in areas of research featuring Hardy´s inequality and related topics. The inequality and its extensions and refinements are not only of intrinsic interest but are indispensable tools in many areas of mathematics and mathematical physics.

Hardy inequalities on domains have a substantial role and this necessitates a detailed investigation of significant geometric properties of a domain and its boundary. Other topics covered in this volume are Hardy- Sobolev-Maz´ya inequalities; inequalities of Hardy-type involving magnetic fields; Hardy, Sobolev and Cwikel-Lieb-Rosenbljum inequalities for Pauli operators; the Rellich inequality.

The Analysis and Geometry of Hardy´s Inequality provides an up-to-date account of research in areas of contemporary interest and would be suitable for a graduate course in mathematics or physics. A good basic knowledge of real and complex analysis is a prerequisite.

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Hardy, Sobolev, and CLR inequalities.- Boundary curvatures and the distance function.- Hardy´s inequality on domains.- Hardy, Sobolev, Maz´ya (HSM) inequalities.- Inequalities and operators involving magnetic elds.- The Rellich inequality.

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"This book is the epitome of classical analysis and has been a staple of those who have wished to learn that art since Cambridge University Press published it in 1934. ... The terseness of the development throughout make this book more suitable for graduate students. All in all, the book under review is a lovely compendium of the utility and power of Hardy´s Inequality." (Jeff Ibbotson, MAA Reviews, maa.org, January, 2016)

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Alexander Balinsky is Professor of Mathematical Physics in the School of Mathematics at Cardiff University. His wide interests include spectral problems for the differential operators of mathematical physics, and currently, the mathematics of image processing, machine learning and data mining.

W. Desmond Evans is now Emeritus Professor in the School of Mathematics at Cardiff University, after spending his working life in Cardiff. He has made contributions to a number of areas of mathematical analysis and mathematical physics, in particular, the spectral analysis of Schrödinger and Dirac operators, non-linear differential operators, functional analysis and operator theory.

Roger T. Lewis is Emeritus Professor in the Department of Mathematics at The University of Alabama at Birmingham, where he has been a faculty member since 1975. His research interest has been mainly in the spectral analysis of the differential operators of mathematical physics, with special attention given to eigenvalue problems of Schrödinger operators and N-body problems of quantum mechanics.

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