Hardy Inequalities on Homogeneous Groups
Hardy Inequalities on Homogeneous Groups

Author: Michael Ruzhansky

Publisher: Springer

Published: 2019-07-02

Total Pages: 579

ISBN-13: 303002895X

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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.

Hardy Inequalities on Homogeneous Groups
Hardy Inequalities on Homogeneous Groups

Author: Durvudkhan Suragan

Publisher:

Published: 2020-10-08

Total Pages: 578

ISBN-13: 9781013273919

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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. This work was published by Saint Philip Street Press pursuant to a Creative Commons license permitting commercial use. All rights not granted by the work's license are retained by the author or authors.

Hardy Inequalities on Homogeneous Groups
Hardy Inequalities on Homogeneous Groups

Author: Michael Ruzhansky

Publisher: Birkhäuser

Published: 2019-07-16

Total Pages: 0

ISBN-13: 9783030028947

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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.

Homogeneous Groups: Hardy Inequalities (Volume 1)
Homogeneous Groups: Hardy Inequalities (Volume 1)

Author: Hart Scott

Publisher: Murphy & Moore Publishing

Published: 2021-11-16

Total Pages: 278

ISBN-13: 9781639873074

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Homogeneous groups are a part of the theories of Lie groups, algebraic groups and topological groups. A homogeneous space for a group G is a non-empty manifold or topological space X on which G acts transitively. The elements of G are known as the symmetries of X. When the group G in question is the automorphism group of the space X, a special case arises. An isometry group, a diffeomorphism group or a homeomorphism group can be called an automorphism group. In this case, X is homogeneous if naturally X looks locally identical at each point, either in the sense of isometry, diffeomorphism or homeomorphism. This book outlines the processes and applications of homogenous groups in detail. It presents this complex subject in the most comprehensible and easy to understand language. This textbook will serve as a valuable source of reference for graduate and post graduate students.

Homogeneous Groups: Hardy Inequalities (Volume 2)
Homogeneous Groups: Hardy Inequalities (Volume 2)

Author: Hart Scott

Publisher: Murphy & Moore Publishing

Published: 2021-11-16

Total Pages: 304

ISBN-13: 9781639873081

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Homogenous groups are part of the theories of Lie groups, algebraic groups and topological groups. A homogeneous space for a group G is a non-empty manifold or topological space X on which G acts transitively. The elements of G are known as the symmetries of X. When the group G in question is the automorphism group of the space X, a special case arises. An isometry group, diffeomorphism group or a homeomorphism group can be called an automorphism group. In this case, X is homogeneous if naturally X looks locally identical at each point, either in the sense of isometry, diffeomorphism or homeomorphism. Thus there is a group action of G on X which can be thought of as preserving some geometric structure on X, and making X into a single G-orbit. This book outlines the processes and applications of homogenous groups in detail. It presents this complex subject in the most comprehensible and easy to understand language. This textbook will serve as a valuable source of reference for graduate and post graduate students.

Hardy Spaces on Homogeneous Groups. (MN-28), Volume 28
Hardy Spaces on Homogeneous Groups. (MN-28), Volume 28

Author: Gerald B. Folland

Publisher: Princeton University Press

Published: 2020-12-08

Total Pages: 302

ISBN-13: 0691222452

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The object of this monograph is to give an exposition of the real-variable theory of Hardy spaces (HP spaces). This theory has attracted considerable attention in recent years because it led to a better understanding in Rn of such related topics as singular integrals, multiplier operators, maximal functions, and real-variable methods generally. Because of its fruitful development, a systematic exposition of some of the main parts of the theory is now desirable. In addition to this exposition, these notes contain a recasting of the theory in the more general setting where the underlying Rn is replaced by a homogeneous group. The justification for this wider scope comes from two sources: 1) the theory of semi-simple Lie groups and symmetric spaces, where such homogeneous groups arise naturally as "boundaries," and 2) certain classes of non-elliptic differential equations (in particular those connected with several complex variables), where the model cases occur on homogeneous groups. The example which has been most widely studied in recent years is that of the Heisenberg group.

Weighted Inequalities of Hardy Type
Weighted Inequalities of Hardy Type

Author: Alois Kufner

Publisher: World Scientific

Published: 2003

Total Pages: 380

ISBN-13: 9789812381958

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Inequalities play an important role in almost all branches of mathematics as well as in other areas of science and engineering. This book surveys the present state of the theory of weighted integral inequalities of Hardy type, including modifications concerning Hardy-Steklov operators, and some basic results about Hardy type inequalities and their limit (Carleman-Knopp type) inequalities. It also describes some rather new fields such as higher order and fractional order Hardy type inequalities and integral inequalities on the cone of monotone functions together with some applications and open problems. The book can serve as a reference and a source of inspiration for researchers working in these and related areas, but could also be used for advanced graduate courses.

The Analysis and Geometry of Hardy's Inequality
The Analysis and Geometry of Hardy's Inequality

Author: Alexander A. Balinsky

Publisher: Springer

Published: 2015-10-20

Total Pages: 277

ISBN-13: 3319228706

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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.

Quantization on Nilpotent Lie Groups
Quantization on Nilpotent Lie Groups

Author: Veronique Fischer

Publisher: Birkhäuser

Published: 2016-03-08

Total Pages: 568

ISBN-13: 3319295586

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This book presents a consistent development of the Kohn-Nirenberg type global quantization theory in the setting of graded nilpotent Lie groups in terms of their representations. It contains a detailed exposition of related background topics on homogeneous Lie groups, nilpotent Lie groups, and the analysis of Rockland operators on graded Lie groups together with their associated Sobolev spaces. For the specific example of the Heisenberg group the theory is illustrated in detail. In addition, the book features a brief account of the corresponding quantization theory in the setting of compact Lie groups. The monograph is the winner of the 2014 Ferran Sunyer i Balaguer Prize.