Introduction 1
Chapter Ⅹ．Existence and Approximation of Solutions of Differential Equations 3
Summary 3
10.1．The Spaces Bp,k 3
10.2．Fundamental Solutions 16
10.3．The Equation P（D）u=f when f∈？′ 29
10.4．Comparison of Differential Operators 32
10.5．Approximation of Solutions of Homogeneous Differential Equations 39
10.6．The Equation P（D）u=f when f is in a Local Space ？′F 41
10.7．The Equation P（D）u=f when f∈？′（X） 45
10.8．The Geometrical Meaning of the Convexity Conditions 50
Notes 58
Chapter Ⅺ．Interior Regularity of Solutions of Differential Equations 60
Summary 60
11.1．Hypoelliptic Operators 61
11.2．Partially Hypoelliptic Operators 69
11.3．Continuation of Differentiability 73
11.4．Estimates for Derivatives of High Order 85
Notes 92
Chapter Ⅻ．The Cauchy and Mixed Problems 94
Summary 94
12.1．The Cauchy Problem for the Wave Equation 96
12.2．The Oscillatory Cauchy Problem for the Wave Equation 104
12.3．Necessary Conditions for Existence and Uniqueness of Solutions to the Cauchy Problem 110
12.4．Properties of Hyperbolic Polynomials 112
12.5．The Cauchy Problem for a Hyperbolic Equation 120
12.6．The Singularities of the Fundamental Solution 125
12.7．A Global Uniqueness Theorem 133
12.8．The Characteristic Cauchy Problem 143
12.9．Mixed Problems 162
Notes 180
Chapter ⅩⅢ．Differential Operators of Constant Strength 182
Summary 182
13.1．Definitions and Basic Properties 182
13.2．Existence Theorems when the Coefficients are Merely Continuous 184
13.3．Existence Theorems when the Coefficients are in C∞ 186
13.4．Hypoellipticity 191
13.5．Global Existence Theorems 194
13.6．Non-uniqueness for the Cauchy Problem 201
Notes 224
Chapter ⅩⅣ．Scattering Theory 225
Summary 225
14.1．Some Function Spaces 227
14.2．Division by Functions with Simple Zeros 232
14.3．The Resolvent of the Unperturbed Operator 237
14.4．Short Range Perturbations 243
14.5．The Boundary Values of the Resolvent and the Point Spectrum 251
14.6．The Distorted Fourier Transforms and the Continuous Spectrum 255
14.7．Absence of Embedded Eigenvalues 264
Notes 268
Chapter ⅩⅤ．Analytic Function Theory and Differential Equations 270
Summary 270
15.1．The Inhomogeneous Cauchy-Riemann Equations 271
15.2．The Fourier-Laplace Transform of Bc2,k（X）when X is Convex 279
15.3．Fourier-Laplace Representation of Solutions of Differential Equations 287
15.4．The Fourier-Laplace Transform of C？（X）when X is Convex 296
Notes 300
Chapter ⅩⅥ．Convolution Equations 302
Summary 302
16.1．Subharmonic Functions 303
16.2．Plurisubharmonic Functions 314
16.3．The Support and Singular Support of a Convolution 319
16.4．The Approximation Theorem 335
16.5．The Inhomogeneous Convolution Equation 341
16.6．Hypoelliptic Convolution Equations 353
16.7．Hyperbolic Convolution Equations 356
Notes 360
Appendix A．Some Algebraic Lemmas 362
A.1．The Zeros of Analytic Functions 362
A.2．Asymptotic Properties of Algebraic Functions of Several Variables 364
Notes 371
Bibliography 373
Index 391
Index of Notation 392