I: Functional analysis.- I. Distribution theory.- 1.0. Introduction.- 1.1. Weak derivatives.- 1.2. Test functions.- 1.3. Definitions and basic properties of distributions.- 1.4. Differentiation of distributions and multiplication by functions.- 1.5. Distributions with compact support.- 1.6. Convolution of distributions.- 1.7. Fourier transforms of distributions.- 1.8. Distributions on a manifold.- II. Some special spaces of distributions.- 2.0. Introduction.- 2.1. Temperate weight functions.- 2.2. The spaces ?p, k.- 2.3. The spaces $$\mathcal{B}_{p,k}^{loc}$$.- 2.4. The spaces ?(s).- 2.5. The spaces ?(m, s).- 2.6. The spaces $$\mathcal{H}_{(s)}^{loc}\left( \Omega \right)$$ when ? is a manifold.- II: Differential operators with constant coefficients.- III. Existence and approximation of solutions of differential equations.- 3.0. Introduction.- 3.1. Existence of fundamental solutions.- 3.2. The equation P (D) u = f when f ? ??.- 3.3. Comparison of differential operators.- 3.4. Approximation of solutions of homogeneous differential equations.- 3.5. The equation P (D) u = f when f is in a local space $$ \subset {\mathcal{D}'_F}$$.- 3.6. The equation P (D) u = f when $$f \in \mathcal{D}'$$.- 3.7. The geometric meaning of P-convexity and strong P-convexity.- 3.8. Systems of differential operators.- IV. Interior regularity of solutions of differential equations.- 4.0. Introduction.- 4.1. Hypoelliptic operators.- 4.2. Partially hypoelliptic operators.- 4.3. Partial hypoellipticity at the boundary.- 4.4. Estimates for derivatives of high order.- V. The Cauchy problem (constant coefficients).- 5.0. Introduction.- 5.1. The classical existence theory for analytic data.- 5.2. The non-uniqueness of the characteristic Cauchy problem.- 5.3. Holmgren’s uniqueness theorem.- 5.4. The necessity of hyperbolicity for the existence of solutions to the noncharacteristic Cauchy problem.- 5.5. Algebraic properties of hyperbolic polynomials.- 5.6. The Cauchy problem for a hyperbolic equation.- 5.7. A global uniqueness theorem.- 5.8. The characteristic Cauchy problem.- III: Differential operators with variable coefficients.- VI. Differential equations which have no solutions.- 6.0. Introduction.- 6.1. Conditions for non-existence.- 6.2. Some properties of the range.- VII. Differential operators of constant strength.- 7.0. Introduction.- 7.1. Definitions and basic properties.- 7.2. Existence theorems when the coefficients are merely continuous.- 7 3 Existence theorems when the coefficients are in C?.- 7.4. Hypoellipticity.- 7.5. The analyticity of the solutions of elliptic equations.- VIII. Differential operators with simple characteristics.- 8.0. Introduction.- 8.1. Necessary conditions for the main estimates.- 8.2. Differential quadratic forms.- 8.3. Estimates for elliptic operators.- 8.4. Estimates for operators with real coefficients.- 8.5. Estimates for principally normal operators.- 8.6. Pseudo-convexity.- 8.7. Estimates, existence and approximation theorems in ?(s).- 8.8. The unique continuation of singularities.- 8.9. The uniqueness of the Cauchy problem.- IX. The Cauchy problem (variable coefficients).- 9.0. Introduction.- 9.1. Preliminary lemmas.- 9.2. The basic L2 estimate.- 9.3. Existence theory for the Cauchy problem.- X. Elliptic boundary problems.- 10.0. Introduction.- 10.1. Definition of elliptic boundary problems.- 10.2. Preliminaries concerning ordinary differential operators.- 10.3. Construction of a parametrix.- 10.4. Local theory of elliptic boundary problems.- 10.5. Elliptic boundary problems in a compact manifold with boundary.- 10.6. Various extensions and remarks.- Appendix. Some algebraic lemmas.- Index of notations.
