**Mathematical Surveys and Monographs**

Volume: 81;
2000;
257 pp;
Softcover

MSC: Primary 35; 42;

**Print ISBN: 978-0-8218-4378-9
Product Code: SURV/81.S**

List Price: $74.00

AMS Member Price: $59.20

MAA Member Price: $66.60

**Electronic ISBN: 978-1-4704-1308-8
Product Code: SURV/81.S.E**

List Price: $69.00

AMS Member Price: $55.20

MAA Member Price: $62.10

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# Tools for PDE: Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials

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*Michael E. Taylor*

This book develops three related tools that are useful in the analysis of
partial differential equations (PDEs), 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. The first chapter studies
classes of pseudodifferential operators whose symbols have a limited
degree of regularity; the second chapter shows how paradifferential
operators yield sharp estimates on the action of various nonlinear
operators on function spaces. The third chapter applies 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 Ricci tensor,
div-curl estimates, and results on propagation of singularities for wave
equations with rough coefficients. The last chapter studies the method of
layer potentials on Lipschitz domains, concentrating on applications to
boundary problems for elliptic PDE with variable coefficients.

#### Readership

Graduate students and research mathematicians interested in partial differential equations.

#### Table of Contents

# Table of Contents

## Tools for PDE: Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials

- Contents vii8 free
- Preface ix10 free
- Chapter 1. Pseudodifferential Operators with Mildly Regular Symbols 112 free
- §1. Spaces of continuous functions 314
- §2. Operator estimates on L[sup(p)], h[sup(1)], and bmo 1728
- §3. Symbol classes and symbol smoothing 3142
- §4. Operator estimates on Sobolev-like spaces 3748
- §5. Operator estimates on spaces C[sup(λ)] 4455
- §6. Products 5465
- §7. Commutator estimates 5869
- §8. Operators with Sobolev coefficients 6172
- §9. Operators with double symbols 6374
- §10. The CRW commutator estimate 7586
- §11. Operators with vmo coefficients 7889
- §12. Estimates on a class of Besov spaces 8293
- §13. Operators with coefficients in a function algebra 8697
- §14. Some BKM-type estimates 8899
- §15. Variations on an estimate of Tumanov 92103
- §16. Estimates on Morrey-type spaces 94105

- Chapter 2. Paradifferential Operators and Nonlinear Estimates 101112
- §1. A product estimate 105116
- §2. A commutator estimate 106117
- §3. Some handy estimates involving maximal functions 108119
- §4. A composition estimate 110121
- §5. More general composition estimate 112123
- §6. Continuity of u [omitted] f(u) on H[sup(1,p)] 113124
- §7. Estimates on F(u) – F(v) 116127
- §8. A pseudodifferential operator estimate 118129
- §9. Paradifferential operators on the spaces C[sup(λ)] 120131
- §A. Paracomposition 125136
- §B. Alinhac's lemma 132143

- Chapter 3. Applications to PDE 135146
- §1. Interior elliptic regularity 137148
- §2. Some natural first-order operators 148159
- §3. Estimates for the Dirichlet problem 155166
- §4. Layer potentials on C[sup(1,ω)] surfaces 159170
- §5. Parametrix estimates and trace asymptotics 173184
- §6. Euler flows on rough planar domains 178189
- §7. Persistence of solutions to semilinear wave equations 183194
- §8. Div-curl estimates 186197
- §9. Harmonic coordinates 194205
- §10. Riemannian manifolds with bounded Ricci tensor 202213
- §11. Propagation of singularities 205216

- Chapter 4. Layer Potentials on Lipschitz Surfaces 217228
- §1. Cauchy kernels on Lipschitz curves 218229
- §2. The method of rotations and extensions to higher dimensions 228239
- §3. The variable-coefficient case 230241
- §4. Boundary integral operators 235246
- §5. The Dirichlet problem on Lipschitz domains 241252
- §A. The Koebe-Bieberbach distortion theorem 246257

- Bibliography 249260
- List of Symbols 255266
- Index 257268 free