Pseudodifferential Operators and Nonlinear PDE

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 L2 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 Lp 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 OPC1Sclm..-4.1 Commutator estimates.-4.2 Operator algebra.-4.3 Gårding inequality.-4.4 C1-paradifferential calculus.- 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.- 9. Extension of the Schauder estimates..-9.1 Nirenberg's refinement.-9.2 Elliptic boundary problems.- A. Function spaces..-A.1 Hölder spaces, Zygmund spaces, and Sobolev spaces.-A.2 Morrey spaces.-A.3 BMO.- B. Sup norm estimates..- C. DeGiorgi-Nash-Moser estimates..-C.2 Hölder continuity.-C.3 Inhomogeneous equations.-C.4 Boundary regularity.- D. Paraproduct estimates..- Index of notation..- References..... Mehr