An Introduction to Sobolev Spaces and Interpolation Spaces

1.Historical background
2.The Lebesgue measure, convolution
3.Smoothing by convolution
4.Truncation, Radon measures, distributions
5.Sobolev spaces, multiplication by smooth functions
6.Density of tensor products, consequences
7.Extending the notion of support
8.Sobolev's embedding theorem, 1 \leq p < N
9.Sobolev's embedding theorem, N \leq p \leq \infty
10.PoincarŽe's inequality.-11.The equivalence lemma, compact embeddings
12.Regularity of the boundary, consequences
13.Traces on the boundary
14.Green's formula.-15.The Fourier transform
16.Traces of Hs(RN)
17.Proving that a point is too small
18.Compact embeddings
19.Lax-Milgram lemma
20.The space H(div; \Omega)
21.Background on interpolation, the complex method
22.Real interpolation: K-method
23.Interpolation of L2 spaces with weights
24.Real interpolation: J-method
25.Interpolation inequalities, the spaces (E0,E1)\theta,1
26.The Lions-Peetre reiteration theorem
27.Maximal functions
28.Bilinear and nonlinear interpolation
29.Obtaining Lp by interpolation, with the exact norm
30.My approach to Sobolev's embedding theorem
31.My generalization of Sobolev's embedding theorem
32.Sobolev's embedding theorem for Besov spaces
33.The Lions-Magenes space H001/2(\Omega )
34.Defining Sobolev spaces and Besov spaces for \Omega
35.Characterization of Ws,p(RN)
36.Characterization of Ws,p (\Omega)
37.Variants with BV spaces
38.Replacing BV by interpolation spaces
39.Shocks for quasi-linear hyperbolic systems
40.Interpolation spaces as trace spaces
41.Duality and compactness for interpolation spaces
42.Miscellaneous questions
43.Biographical information
44.Abbreviations and mathematical notation
- References
- Index

Luc Tartar studied at Ecole Polytechnique in Paris, France, 1965-1967, where he was taught by Laurent Schwartz and Jacques-Louis Lions in mathematics, and by Jean Mandel in continuum mechanics.