1. Historical Background ....................................... 1
2. The Lebesgue Measure, Convolution ........................... 9
3. Smoothing by Convolution ................................... 15
4. Truncation; Radon Measures; Distributions .................. 17
5. Sobolev Spaces; Multiplication by Smooth Functions ......... 21
6. Density of Tensor Products; Consequences ................... 27
7. Extending the Notion of Support ............................ 33
8. Sobolev's Embedding Theorem, 1 ≤ p < N ..................... 37
9. Sobolev's Embedding Theorem, N ≤ p ≤ ∞ ..................... 43
10. Poincare's Inequality ...................................... 49
11. The Equivalence Lemma; Compact Embeddings .................. 53
12. Regularity of the Boundary; Consequences ................... 59
13. Traces on the Boundary ..................................... 65
14. Green's Formula ............................................ 69
15. The Fourier Transform ...................................... 73
16. Traces of HB(RN) ........................................... 81
17. Proving that a Point is too Small .......................... 85
18. Compact Embeddings ......................................... 89
19. Lax-Milgram Lemma .......................................... 93
20. The Space H(div; Ω) ........................................ 99
21. Background on Interpolation; the Complex Method ........... 103
22. Real Interpolation; K-Method .............................. 109
23. Interpolation of L2 Spaces with Weights ................... 115
24. Real Interpolation; J-Method .............................. 119
25. Interpolation Inequalities, the Spaces (E0,E1)θ,1 .......... 123
26. The Lions-Peetre Reiteration Theorem ...................... 127
27. Maximal Functions ......................................... 131
28. Bilinear and Nonlinear Interpolation ...................... 137
29. Obtaining Lp by Interpolation, with the Exact Norm ........ 141
30. My Approach to Sobolev's Embedding Theorem ................ 145
31. My Generalization of Sobolev's Embedding Theorem .......... 149
32. Sobolev's Embedding Theorem for Besov Spaces .............. 155
33. The Lions-Magenes Space H001/2(Ω) .......................... 159
34. Denning Sobolev Spaces and Besov Spaces for Ω ............. 163
35. Characterization of Ws,p(RN) ............................... 165
36. Characterization of Ws,p(Ω)................................. 169
37. Variants with BV Spaces ................................... 173
38. Replacing BV by Interpolation Spaces ...................... 177
39. Shocks for Quasi-Linear Hyperbolic Systems ................ 183
40. Interpolation Spaces as Trace Spaces ...................... 191
41. Duality and Compactness for Interpolation Spaces .......... 195
42. Miscellaneous Questions ................................... 199
43. Biographical Information .................................. 205
44. Abbreviations and Mathematical Notation ................... 209
References .................................................... 213
Index ......................................................... 215
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