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..