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

"This book is both an excellent introduction for those learning about heat operators for the first time, and a reference work for the mathematician searching for information. The author has presented an especially lucid exposition of the subject." - Alan McIntosh, Australian National University; "This book contains very interesting material, starting with the basics and progressing to lively trends of current research." - Thierry Coulhon, Cergy-Pontoise University"

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

"This book is both an excellent introduction for those learning about heat operators for the first time, and a reference work for the mathematician searching for information. The author has presented an especially lucid exposition of the subject." - Alan McIntosh, Australian National University; "This book contains very interesting material, starting with the basics and progressing to lively trends of current research." - Thierry Coulhon, Cergy-Pontoise University"

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

"This book is both an excellent introduction for those learning about heat operators for the first time, and a reference work for the mathematician searching for information. The author has presented an especially lucid exposition of the subject." - Alan McIntosh, Australian National University; "This book contains very interesting material, starting with the basics and progressing to lively trends of current research." - Thierry Coulhon, Cergy-Pontoise University"