## Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials

This book is number 81 in the AMS Series, Mathematical Surveys and Monographs. It develops three related tools that are useful in the analysis of partial differential equations, arising from the classical study of singular integral operators: pseudodifferential operators, paradifferential operators, and layer potentials.

A theme running throughout the work is the treatment of PDE in the presence of relatively little regularity. In the first chapter we study classes of pseudodifferential operators whose symbols have a limited degree of regularity. In the second chapter we show how paradifferential operators yield sharp estimates on various nonlinear operators on function spaces. In Chapter 3 we apply this material to an assortment of results in PDE, including regularity results for elliptic PDE with rough coefficients, planar fluid flows on rough domains, estimates on Riemannian manifolds given weak bounds on the Ricci tensor, div-curl estimates, and results on propagation of singularities for wave equations with rough coefficients. Chapter 4 studies the method of layer potentials on Lipschitz domains, concentrating on applications to boundary problems for elliptic PDE with variable coefficients.

To some degree this monograph is a companion to Pseudodifferential Operators and Nonlinear PDE

## Contents

- Chapter 1. Pseudodifferential operators with mildly regular symbols
- Spaces of continuous functions
- Operator estimates on L
^{p}, h^{1}, and bmo - Symbol classes and symbol smoothing
- Operator estimates on Sobolev-like spaces
- Operator estimates on spaces C
^{(l)} - Products
- Commutator estimates
- Operators with Sobolev coefficients
- Operators with double symbols
- The CRW commutator estimate
- Operators with vmo coefficients
- Estimates on a class of Besov spaces
- Operators with coefficients in a function algebra
- Some BKM-type estimates
- Variations on an estimate of Tumanov
- Estimates on Morrey-type spaces

- Chapter 2. Paradifferential operators and nonlinear estimates
- A product estimate
- A commutator estimate
- Some handy estimates involving maximal functions
- A composition estimate
- More general composition estimates
- Continuity of f(u) on H
^{1,p} - Estimates on F(u)-F(v)
- A pseudodifferential operator estimate
- Paradifferential operators on the spaces C
^{(l)}

- Chapter 3. Applications to PDE
- Interior elliptic regularity
- Some natural first-order operators
- Estimates for the Dirichlet problem
- Layer potentials on C
^{1,w}surfaces - Parametrix estimates and trace asymptotics
- Euler flows on rough planar domains
- Persistence of solutions to semilinear wave equations
- Div-curl estimates
- Harmonic coordinates
- Riemannian manifolds with bounded Ricci tensor
- Propagation of singularities

- Chapter 4. Layer potentials on Lipschitz surfaces
- Cauchy kernels on Lipschitz curves
- The method of rotations and extensions to higher dimensions
- The variable-coefficient case
- Boundary integral operators
- The Dirichlet problem on Lipschitz domains