Inequalities

Selecta of Elliott H. Lieb

Springer-Verlag Berlin Heidelberg GmbH

ELLIOTT H. LIEB

Inequalities Selecta of Elliott H. Lieb

Edited by M. Loss and M.B. Ruskai

Springer

Professor Elliott H. Lieb Jadwin Hall

Departments of Mathematics and Physics Princeton University

P.O. Box 708 Princeton, New Jersey 08544-0708, USA

Professor Michael Loss School of Mathematics

Georgia Tech Atlanta, GA 30332-0160, USA

Professor Mary Beth Ruskai Department of Mathematics

University of Massachusetts Lowell Lowell, MA 01854, USA

Library of Congress Cataloging-in-Publication Data

Lieb, Elliott H. Inequalities : selecta of Elliott H. Lieb / edited by M. Loss and M.B. Ruskai. p. cm.

Includes bibliographical references. ISBN 978-3-642-62758-3 ISBN 978-3-642-55925-9 (eBook) DOI 10.1007/978-3-642-55925-9

1. Inequalities (Mathematics) 1. Loss, 1954-II. Ruskai, Mary Beth. III. Title. QA295 .L54 2002 515'.26--dc21 2002021784

QC173.4.T48 L54 2001 539'.1-dc21 2001041096

First Edition 2002. Corrected Second Printing 2003

ISBN 978-3-642-62758-3

This work is subject to copyright. AlI rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on micro­

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Preface

Elliott Lieb made seminal contributions to physics and mathematics. The former are partially collected in the volume "The Stability of Matter: From Atoms to Stars" (Selecta of Elliott Lieb), now in its third edition, which contains some of his papers on the structure of matter, such as its stability and the existence of thermodynamic functions. This new volume is a selection of his contributions to analysis, in particular his work on inequalities.

There are several reasons for publishing this collection of his work. Many of Lieb's results have a substantial impact on analysis, such as his work with Bras­camp that determined the sharp constant in Young's inequality. Another example is his work with various collaborators on rearrangement inequalities, which is an area now fully recognized as a very valuable part of analysis.

A second important consideration is the long shelflife ofLieb's work. For ex­ample, the paper 'Convex Trace Functions and the Wigner-Yanase-Dyson Con­jecture' (No 11.3 of the present volume) has been cited in a variety of contexts in the nearly 30 years since its publication, most recently in connection with the theory of quantum computing.

Lieb's work is a fortunate exception to the rule that research papers are usually only accessible to experts. The reader will find that a good background in real analysis and linear algebra is sufficient for understanding and even mastering the content of most of his papers. The vitality of his papers springs from the ideas and not from the technical complexities.

Last but certainly not least is the relevance of Lieb' s papers to physics. This is demonstrated by the now famous Lieb-Thirring inequalities. A nice example is also his work with Brascamp about log-concave functions and their application to the one dimensional quantum Wigner crystal. It is satisfying to see how, starting with the simple idea of convexity, a complete understanding of a nontrivial piece of physics emerges.

The papers are grouped around physical and mathematical ideas. We have added commentaries that serve to introduce the papers. Some of these explain aspects of the history of the problem or of the paper, and others point towards further developments. We have chosen to comment mostly on those papers in ar­eas where we have some research expertise. We hope that the reader enjoys this collection of Lieb's papers as much as we do.

Our thanks go to all the publishers who generously supplied the papers free of charge, to the staff of Springer Verlag, and especially to Wolf BeiglbOck for his support and his patience.

Atlanta and Lowell, 2002 Michael Loss and Mary Beth Ruskai

v

Contents

Commentaries

Part I. Inequalities Related to Statistical Mechanics and Condensed Matter

Ll Theory of Ferromagnetism and the Ordering of Electronic Energy Levels (with D.C. Mattis) .......... 33

1.2 Ordering Energy Levels of Interacting Spin Systems (with D.C. Mattis) .................. 43

1.3 Entropy Inequalities (with H. Araki) ........ 47 1.4 A Fundamental Property of Quantum-Mechanical Entropy

(with M.B. Ruskai) . . . . . . . . . . . . . . . . . . . . . 59 1.5 Proof of the Strong Subadditivity of Quantum-Mechanical Entropy

(with M.B. Ruskai) ................... 63 1.6 Some Convexity and Subadditivity Properties of Entropy 67 1.7 A Refinement of Simon's Correlation Inequality 81 I.8 Two Theorems on the Hubbard Model . . . . . . . . . . 91 1.9 Magnetic Properties of Some Itinerant-Electron Systems at T > 0

(with M. Aizenman) ..................... 95

Part II. Matrix Inequalities and Combinatorics

ILl Proofs of Some Conjectures on Permanents . . . 101 11.2 Concavity Properties and a Generating Function

for Stirling Numbers ............... 109 11.3 Convex Trace Functions and the Wigner-Yanase-Dyson Conjecture 113 11.4 Some Operator Inequalities of the Schwarz Type

(with M.B. Ruskai) ......................... 135 11.5 Inequalities for Some Operator and Matrix Functions . . . . . . . . 141 11.6 Positive Linear Maps Which Are Order Bounded on C* Subalgebras

(with M. Aizenman and E.B. Davies) ............ 147 11.7 Optimal Hypercontractivity for Fermi Fields and Related

Non-Commutative Integration Inequalities (with E. Carlen) 151 11.8 Sharp Uniform Convexity and Smoothness Inequalities

for Trace Norms (with K. Ball and E. Carlen) . . . . . . . . 171 11.9 A Minkowski Type Trace Inequality and Strong Subadditivity

of Quantum Entropy (with E. Carlen) ............. 191

VII

Part III. Inequalities Related to the Stability of Matter

III. 1 Inequalities for the Moments of the Eigenvalues of the Schr6dinger Hamiltonian and Their Relation to Sobolev Inequalities (with W. Thirring) ..................... 203

IlL2 On Semi-Classical Bounds for Eigenvalues of Schr6dinger Operators (with M. Aizenman) .............. 239

IlI.3 The Number of Bound States of One-Body SchrOdinger Operators and the Weyl Problem .................. 243

IlIA Improved Lower Bound on the Indirect Coulomb Energy (with S. Oxford) .............. 255

IlL5 Density Functionals for Coulomb Systems . 269 III.6 On Characteristic Exponents in Turbulence 305 IlI.7 Baryon Mass Inequalities in Quark Models 313 III. 8 Kinetic Energy Bounds and Their Application to the Stability

of Matter ............................ 317 IlL 9 A Sharp Bound for an Eigenvalue Moment of the One-Dimensional

SchrOdinger Operator (with D. Hundertmark and L.E. Thomas) 329

Part IV. Coherent States

IV.1 The Classical Limit of Quantum Spin Systems ........... 345 IV.2 Proof of an Entropy Conjecture of Wehr I . . . . . . . . . . . 359 IV.3 Quantum Coherent Operators: A Generalization of Coherent States

(with J.P. Solovej) ..................... 367 IVA Coherent States as a Tool for Obtaining Rigorous Bounds 377

Part V. Brunn-Minkowski Inequality and Rearrangements

V.l A General Rearrangement Inequality for Multiple Integrals (with H.J. Brascamp and J.M. Luttinger) .......... 391

V.2 Some Inequalities for Gaussian Measures and the Long-Range Order of the One-Dimensional Plasma (with H.J. Brascamp) 403

V.3 Best Constants in Young's Inequality, Its Converse and Its Generalization to More than Three Functions (with H.J. Brascamp) 417

VA On Extensions of the Brunn-Minkowski and Prekopa-Leindler Theorems, Including Inequalities for Log Concave Functions and with an Application to the Diffusion Equation (with R.J. Brascamp) ................. 441

V.5 Existence and Uniqueness of the Minimizing Solution ofChoquard's Nonlinear Equation . . . . . . . . . . . 465

V.6 Symmetric Decreasing Rearrangement Can Be Discontinuous (with F. Almgren) ....................... 479

V.7 The (Non) Continuity of Symmetric Decreasing Rearrangement (with F. Almgren) ...................... 483

V.8 On the Case of Equality in the Brunn-Minkowski Inequality for Capacity (with L. Cafarelli and D. Jerison) ....... 497

VIII

VI. 1

Part VI. General Analysis

An U Bound for the Riesz and Bessel Potentials of Orthonormal Functions ........... .

VI.2 A Relation Between Pointwise Convergence of Functions 515

and Convergence of Functionals (with H. Brezis) . 523 VI.3 Sharp Constants in the Hardy-Littlewood-Sobolev

and Related Inequalities .............. 529 VIA On the Lowest Eigenvalue of the Laplacian for the Intersection

of Two Domains ....................... 555 VI. 5 Minimum Action Solutions of Some Vector Field Equations

(with H. Brezis) . . . . . . . . . . . . . . . . . . . . . . . . 563 VI.6 Sobolev Inequalities with Remainder Terms (with H. Brezis) 581 VI.7 Gaussian Kernels Have Only Gaussian Maximizers ......... 595 VI.8 Integral Bounds for Radar Ambiguity Functions

and Wigner Distributions . . . . . . . . . . . . . . . . . . . 625

Part VII. Inequalities Related to Harmonic Maps

VII. 1 Estimations d'energie pour des applications de R3 it valeurs dans S2 (with H. Brezis and J-M. Coron) ........ 633

VII.2 Singularities of Energy Minimizing Maps from the Ball to the Sphere (with F. Almgren) ............. 637

VII.3 Co-area, Liquid Crystals, and Minimal Surfaces (with F. Almgren and W. Browder) ........... 641

VIlA Counting Singularities in Liquid Crystals (with F. Almgren) 663 VII.5 Symmetry of the Ginzburg-Landau Minimizer in a Disc

(with M. Loss) ........................ 679

Publications of Elliott H. Lieb 695

IX