viii Contents
6.1. General form of a distribution with support at 0 20
6.2. Distributions with compact supports 22
§7. The convolution of distributions 24
7.1. Convolution of f D and ϕ C0

24
7.2. Convolution of f D and g E 26
7.3. Direct product of distributions 27
7.4. Partial hypoellipticity 28
§8. Problems 30
Chapter II. Fourier Transforms 33
§9. Tempered distributions 33
9.1. General form of a tempered distribution 35
§10. Fourier transforms of tempered distributions 37
10.1. Fourier transforms of functions in S 38
10.2. Fourier transform of tempered distributions 39
10.3. Generalization of Liouville’s theorem 41
§11. Fourier transforms of distributions with compact supports 42
§12. Fourier transforms of convolutions 45
§13. Sobolev spaces 46
13.1. Density of C0
∞(Rn)
in
Hs(Rn)
49
13.2. Multiplication by a(x) S 50
13.3. Sobolev’s embedding theorem 51
13.4. An equivalent norm for noninteger 52
13.5. Restrictions to hyperplanes (traces) 53
13.6. Duality of Sobolev spaces 54
13.7. Invariance of
Hs(Rn)
under changes of variables 55
§14. Singular supports and wave front sets of distributions 57
14.1. Products of distributions 61
14.2. Restrictions of distributions to a surface 63
§15. Problems 65
Chapter III. Applications of Distributions to Partial Differential
Equations 69
§16. Partial differential equations with constant coefficients 69
16.1. The heat equation 70
16.2. The Schr¨ odinger equation 72
16.3. The wave equation 73
16.4. Fundamental solutions for the wave equations 74
16.5. The Laplace equation 78
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