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Preface ix Notation xiii Chapter 1. SESQUILINEAR FORMS, ASSOCIATEDOPERATORS, AND SEMIGROUPS 1 1.1 Bounded sesquilinear forms 1 1.2 Unboundedsesquilinear forms and their associated operators 3 1.3 Semigroups andunbounded operators 18 1.4 Semigroups associated with sesquilinear forms 291.5 Correspondence between forms, operators, and semigroups 38 Chapter 2.CONTRACTIVITY PROPERTIES 43 2.1 Invariance of closed convex sets 44 2.2Positive and Lp-contractive semigroups 49 2.3 Domination of semigroups 582.4 Operations on the form-domain 64 2.5 Semigroups acting on vector-valuedfunctions 68 2.6 Sesquilinear forms with nondense domains 74 Chapter 3.INEQUALITIES FOR SUB-MARKOVIAN SEMIGROUPS 79 3.1 Sub-Markovian semigroupsand Kato type inequalities 79 3.2 Further inequalities and the correspondingdomain in Lp 88 3.3 Lp-holomorphy of sub-Markovian semigroups 95 Chapter 4.UNIFORMLY ELLIPTIC OPERATORS ON DOMAINS 99 4.1 Examples of boundaryconditions 99 4.2 Positivity and irreducibility 103 4.3 L1-contractivity107 4.4 The conservation property 120 4.5 Domination 125 4.6Lp-contractivity for 1 134 4.7 Operators with unbounded coefficients 137Chapter 5. DEGENERATE-ELLIPTIC OPERATORS 143 5.1 Symmetricdegenerate-elliptic operators 144 5.2 Operators with terms of order 1 145Chapter 6. GAUSSIAN UPPER BOUNDS FOR HEAT KERNELS 155 6.1 Heat kernelbounds, Sobolev, Nash, and Gagliardo-Nirenberg inequalities 155 6.2Holder-continuity estimates of the heat kernel 160 6.3 Gaussian upper bounds163 6.4 Sharper Gaussian upper bounds 174 6.5 Gaussian bounds for complextime and Lp-analyticity 180 6.6 Weighted gradient estimates 185 Chapter 7.GAUSSIAN UPPER BOUNDS AND Lp-SPECTRAL THEORY 193 7.1 Lp-bounds andholomorphy 196 7.2 Lp-spectral independence 204 7.3 Riesz means andregularization of the Schrodinger group 208 7.4 Lp-estimates for waveequations 214 7.5 Singular integral operators on irregular domains 228 7.6Spectral multipliers 235 7.7 Riesz transforms associated with uniformlyelliptic operators 240 7.8 Gaussian lower bounds 245 Chapter 8. A REVIEW OFTHE KATO SQUARE ROOT PROBLEM 253 8.1 The problem in the abstract setting 2538.2 The Kato square root problem for elliptic operators 257 8.3 Someconsequences 261 Bibliography 265 Index 283