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