This is the first comprehensive reference published on heat equations associated with non self-adjoint uniformly elliptic operators. The author provides introductory materials for those unfamiliar with the underlying mathematics and background needed to understand the properties of heat equations. He then treats Lp properties of solutions to a wide class of heat equations that have been developed over the last fifteen years. These primarily concern the interplay of heat equations in functional analysis, spectral theory and mathematical physics. This book addresses new developments and applications of Gaussian upper bounds to spectral theory. In particular, it shows how such bounds can be used in order to prove Lp estimates for heat, Schrodinger, and wave type equations. A significant part of the results have been proved during the last decade. The book will appeal to researchers in applied mathematics and functional analysis, and to graduate students who require an introductory text to sesquilinear form techniques, semigroups generated by second order elliptic operators in divergence form, heat kernel bounds, and their applications.
It will also be of value to mathematical physicists. The author supplies readers with several references for the few standard results that are stated without proofs.
El-Maati Ouhabaz is Professor of Analysis and Geometry at Universite Bordeaux 1
Preface ix Notation xiii Chapter 1. SESQUILINEAR FORMS, ASSOCIATED OPERATORS, AND SEMIGROUPS 1 1.1 Bounded sesquilinear forms 1 1.2 Unbounded sesquilinear forms and their associated operators 3 1.3 Semigroups and unbounded operators 18 1.4 Semigroups associated with sesquilinear forms 29 1.5 Correspondence between forms, operators, and semigroups 38 Chapter 2. CONTRACTIVITY PROPERTIES 43 2.1 Invariance of closed convex sets 44 2.2 Positive and Lp-contractive semigroups 49 2.3 Domination of semigroups 58 2.4 Operations on the form-domain 64 2.5 Semigroups acting on vector-valued functions 68 2.6 Sesquilinear forms with nondense domains 74 Chapter 3. INEQUALITIES FOR SUB-MARKOVIAN SEMIGROUPS 79 3.1 Sub-Markovian semigroups and Kato type inequalities 79 3.2 Further inequalities and the corresponding domain in Lp 88 3.3 Lp-holomorphy of sub-Markovian semigroups 95 Chapter 4. UNIFORMLY ELLIPTIC OPERATORS ON DOMAINS 99 4.1 Examples of boundary conditions 99 4.2 Positivity and irreducibility 103 4.3 L1-contractivity 107 4.4 The conservation property 120 4.5 Domination 125 4.6 Lp-contractivity for 1 134 4.7 Operators with unbounded coefficients 137 Chapter 5. DEGENERATE-ELLIPTIC OPERATORS 143 5.1 Symmetric degenerate-elliptic operators 144 5.2 Operators with terms of order 1 145 Chapter 6. GAUSSIAN UPPER BOUNDS FOR HEAT KERNELS 155 6.1 Heat kernel bounds, Sobolev, Nash, and Gagliardo-Nirenberg inequalities 155 6.2 Holder-continuity estimates of the heat kernel 160 6.3 Gaussian upper bounds 163 6.4 Sharper Gaussian upper bounds 174 6.5 Gaussian bounds for complex time and Lp-analyticity 180 6.6 Weighted gradient estimates 185 Chapter 7. GAUSSIAN UPPER BOUNDS AND Lp-SPECTRAL THEORY 193 7.1 Lp-bounds and holomorphy 196 7.2 Lp-spectral independence 204 7.3 Riesz means and regularization of the Schrodinger group 208 7.4 Lp-estimates for wave equations 214 7.5 Singular integral operators on irregular domains 228 7.6 Spectral multipliers 235 7.7 Riesz transforms associated with uniformly elliptic operators 240 7.8 Gaussian lower bounds 245 Chapter 8. A REVIEW OF THE KATO SQUARE ROOT PROBLEM 253 8.1 The problem in the abstract setting 253 8.2 The Kato square root problem for elliptic operators 257 8.3 Some consequences 261 Bibliography 265 Index 283