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.Poincar´e’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.