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-
  1. The Fourier integral representation and symbol classes
  2. Schwartz kernels of pseudodifferential operators
  3. Adjoints and products
  4. Elliptic operators and parametrices
  5. L² estimates
  6. Garding’s inequality
  7. The sharp Garding inequality
  8. Hyperbolic evolution equations
  9. Egorov’s theorem
  10. Microlocal regularity
  11. L^p estimates
  12. Operators on manifolds
• Chapter 1. Symbols with limited smoothness
  1. Symbol classes
  2. Some simple elliptic regularity theorems
  3. Symbol smoothing
• Chapter 2. Operator estimates and elliptic regularity
  1. Bounds on operators with nonregular symbols
  2. Further elliptic regularity theorems
  3. Adjoints
  4. Sharp Garding inequality
• Chapter 3. Paradifferential operators
  1. Composition and paraproducts
  2. Various forms of paraproduct
  3. Nonlinear PDE and paradifferential operators
  4. Operator algebra
  5. Product estimates
  6. Commutator estimates
• Chapter 4. Calculus for OPC¹S^m_cl
  1. Commutator estimates
  2. Operator algebra
  3. Garding inequality
  4. C¹-paradifferential calculus
• Chapter 5. Nonlinear hyperbolic systems
  1. Quasilinear symmetric hyperbolic systems
  2. Symmetrizable hyperbolic systems
  3. Higher order hyperbolic equations
  4. Completely nonlinear hyperbolic systems
• Chapter 6. Propagation of singularities
  1. Propagation of singularities
  2. Nonlinear formation of singularities
  3. Egorov’s theorem
• Chapter 7. Nonlinear parabolic systems
  1. Strongly parabolic quasilinear systems
  2. Petrowski parabolic quasilinear systems
  3. Sharper estimates
  4. Semilinear parabolic systems
• Chapter 8. Nonlinear elliptic boundary problems
  1. Second order elliptic equations
  2. Quasilinear elliptic equations
  3. Interface with DeGiorgi-Nash-Moser theory
• Chapter 9. Extension of the Schauder estimates
  1. Nirenberg’s refinement
  2. Elliptic boundary problems
• Appendix A. Function spaces
  1. Holder spaces, Zygmund spaces, and Sobolev spaces
  2. Morrey spaces
  3. BMO
• Appendix B. Sup norm estimates
  1. L^oo estimates on pseudodifferential operators
  2. The spaces C^r_#
• Appendix C. DeGiorgi-Nash-Moser estimates
  1. Moser iteration and L^oo estimates
  2. Holder continuity
  3. Inhomogeneous equations
  4. Boundary regularity
• Paraproduct estimates