Pseudodifferential Operators And Nonlinear PDE

CONTENTS Introduction. 0. Pseudodifferential operators and linear PDE. §0.1 The Fourier integral representation and symbol classes §0.2 Schwartz kernels of pseudodifferential operators §0.3 Adjoints and products §0.4 Elliptic operators and parametrices §0.5 L 2 estimates §0.6 Gårding's inequality §0.7 The sharp Gårding inequality §0.8 Hyperbolic evolution equations §0.9 Egorov's theorem §0.10 Microlocal regularity §0.11 L p estimates §0.12 Operators on manifolds 1. Symbols with limited smoothness. §1.1 Symbol classes §1.2 Some simple elliptic regularity theorems §1.3 Symbol smoothing 2. Operator estimates and elliptic regularity. §2.1 Bounds for operators with nonregular symbols §2.2 Further elliptic regularity theorems §2.3 Adjoints §2.4 Sharp Gårding inequality 3. Paradifferential operators. §3.1 Composition and paraproducts §3.2 Various forms of paraproduct §3.3 Nonlinear PDE and paradifferential operators §3.4 Operator algebra §3.5 Product estimates §3.6 Commutator estimates 4. Calculus for OP C 1 S m cl. §4.1 Commutator estimates §4.2 Operator algebra §4.3 Gårding inequality §4.4 C 1-paradifferential calculus 3 5. Nonlinear hyperbolic systems. §5.1 Quasilinear symmetric hyperbolic systems §5.2 Symmetrizable hyperbolic systems §5.3 Higher order hyperbolic equations §5.4 Completely nonlinear hyperbolic systems 6. Propagation of singularities. §6.1 Propagation of singularities §6.2 Nonlinear formation of singularities §6.3 Egorov's theorem 7. Nonlinear parabolic systems. §7.1 Strongly parabolic quasilinear systems §7.2 Petrowski parabolic quasilinear systems §7.3 Sharper estimates §7.4 Semilinear parabolic systems 8. Nonlinear elliptic boundary problems. §8.1 Second order elliptic equations §8.2 Quasilinear elliptic equations §8.3 Interface with DeGiorgi-Nash-Moser theory A. Function spaces. §A.1 Hölder spaces, Zygmund spaces, and Sobolev spaces §A.2 Morrey spaces §A.3 BMO B. Sup norm estimates. §B.1 L ∞ estimates on pseudodifferential operators §B.2 The spaces C r # = B r ∞,1 C. DeGiorgi-Nash-Moser estimates. §C.1 Moser iteration and L ∞ estimates §C.2 Hölder continuity §C.3 Inhomogeneous equations §C.4 Boundary regularity D. Paraproduct estimates. Index of notation. References.