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=9. Sobolev's Embedding Theorem, N=p=8

10. Poincaramp;#x00E9;'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;O)

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)?,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 H_{00}^{1/2} ( Omega)

34. Defining Sobolev Spaces and Besov Spaces for O

35. Characterization of Ws,p(RN)

36. Characterization of Ws,p(O)

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

DRM-restrictions

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Keywords: MATHEMATICS / General MAT000000