Tartar L. An Introduction to Sobolev Spaces and Interpolation Spaces

Springer-Verlag. Berlin Heidelberg. 2007. — 219 p.Appears parallel to the conference in honor of Luc Tartar on the occasion of his 60th birthday held in Paris, July 2-6, 2007 at the CMAP of the Ecole Polytechnique.During his long career, Luc Tartar had not written a book until 2006 when the new series Lecture Notes of the Unione Matematica Italiana started publication. This is his second volume in the new series and a third is in preparation.Historical Background.
The Lebesgue Measure, Convolution.
Smoothing by Convolution.
Truncation; Radon Measures; Distributions.
Sobolev Spaces; Multiplication by Smooth Functions.
Density of Tensor Products; Consequences.
Extending the Notion of Support.
Sobolev’s Embedding Theorem, 1 ≤ p < N.
Sobolev’s Embedding Theorem, N ≤ p≤∞.
Poincar´e’s Inequality.
The Equivalence Lemma; Compact Embeddings.
Regularity of the Boundary; Consequences.
Traces on the Boundary.
Green’s Formula.
The Fourier Transform.
Traces of Hs(RN).
Proving that a Point is too Small.
Compact Embeddings.
Lax – Milgram Lemma.
The Space H(div; Ω).
Background on Interpolation; the Complex Method.
Real Interpolation; K-Method.
Interpolation of L2 Spaces with Weights.
Real Interpolation; J-Method.
Interpolation Inequalities, the Spaces (E0, E1)θ,1.
The Lions – Peetre Reiteration Theorem.
Maximal Functions.
Bilinear and Nonlinear Interpolation.
Obtaining Lp by Interpolation, with the Exact Norm.
My Approach to Sobolev’s Embedding Theorem.
My Generalization of Sobolev’s Embedding Theorem.
Sobolev’s Embedding Theorem for Besov Spaces.
The Lions – Magenes Space H1/200 (Ω).
Defining Sobolev Spaces and Besov Spaces for Ω.
Characterization of Ws,p(RN).
Characterization of Ws,p(Ω).
Variants with BV Spaces.
Replacing BV by Interpolation Spaces.
Shocks for Quasi-Linear Hyperbolic Systems.
Interpolation Spaces as Trace Spaces.
Duality and Compactness for Interpolation Spaces.
Miscellaneous Questions.
Biographical Information.
Abbreviations and Mathematical Notation.