This book is number 100 in the Birkhauser Series, Progress in Mathematics. It deals with the use of pseudodifferential operators as a tool in nonlinear PDE.

One goal has been to build a bridge between two approaches that have been used in a number of works, one being the theory of paradifferential operators, introduced by J.-M. Bony, the other the study of pseudodifferential operators whose symbols have limited regularity.

To some degree the monograph Tools for PDE is a companion to this volume.

## Contents

- Chapter 0. Pseudodifferential operators and linear PDE-
- The Fourier integral representation and symbol classes
- Schwartz kernels of pseudodifferential operators
- Adjoints and products
- Elliptic operators and parametrices
- L
^{2}estimates - Garding’s inequality
- The sharp Garding inequality
- Hyperbolic evolution equations
- Egorov’s theorem
- Microlocal regularity
- L
^{p}estimates - Operators on manifolds

- Chapter 1. Symbols with limited smoothness
- Symbol classes
- Some simple elliptic regularity theorems
- Symbol smoothing

- Chapter 2. Operator estimates and elliptic regularity
- Bounds on operators with nonregular symbols
- Further elliptic regularity theorems
- Adjoints
- Sharp Garding inequality

- Chapter 3. Paradifferential operators
- Composition and paraproducts
- Various forms of paraproduct
- Nonlinear PDE and paradifferential operators
- Operator algebra
- Product estimates
- Commutator estimates

- Chapter 4. Calculus for OPC
^{1}S^{m}_{cl}- Commutator estimates
- Operator algebra
- Garding inequality
- C
^{1}-paradifferential calculus

- Chapter 5. Nonlinear hyperbolic systems
- Quasilinear symmetric hyperbolic systems
- Symmetrizable hyperbolic systems
- Higher order hyperbolic equations
- Completely nonlinear hyperbolic systems

- Chapter 6. Propagation of singularities
- Propagation of singularities
- Nonlinear formation of singularities
- Egorov’s theorem

- Chapter 7. Nonlinear parabolic systems
- Strongly parabolic quasilinear systems
- Petrowski parabolic quasilinear systems
- Sharper estimates
- Semilinear parabolic systems

- Chapter 8. Nonlinear elliptic boundary problems
- Second order elliptic equations
- Quasilinear elliptic equations
- Interface with DeGiorgi-Nash-Moser theory

- Chapter 9. Extension of the Schauder estimates
- Nirenberg’s refinement
- Elliptic boundary problems

- Appendix A. Function spaces
- Holder spaces, Zygmund spaces, and Sobolev spaces
- Morrey spaces
- BMO

- Appendix B. Sup norm estimates
- L
^{oo}estimates on pseudodifferential operators - The spaces C
^{r}_{#}

- L
- Appendix C. DeGiorgi-Nash-Moser estimates
- Moser iteration and L
^{oo}estimates - Holder continuity
- Inhomogeneous equations
- Boundary regularity

- Moser iteration and L
- Paraproduct estimates