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Sobolev spaces. (English) Zbl 1044.46031
Johnson, W. B. (ed.) et al., Handbook of the geometry of Banach spaces. Volume 2. Amsterdam: North-Holland (ISBN 0-444-51305-1/hbk). 1361-1423 (2003).
This is an excellent survey article on Sobolev spaces including the proofs of most of the theorems presented here. The paper is divided into eight sections. The first section introduces the classical definitions: Let \( \Omega \subset {\mathbb R}^n \) be open and let \(D^{ \alpha}f\) be the \( \alpha\)-th distributional partial derivative for a function \(f: \Omega \rightarrow {\mathbb K}\) (i.e., \({\mathbb R}\) or \({\mathbb C}\)) where \( \alpha\) is a multiindex. Then the authors study Sobolev spaces \[ L^p_{(k)}( \Omega) = \{ f: \Omega \rightarrow {\mathbb K} : D^{ \alpha}f {\text{ exists and }} D^{ \alpha}f \in L^p( \Omega) {\text{ for }} | \alpha | \leq k \} \] endowed with the norm \[ | | f| | = \begin{cases} \left( \sum_{| \alpha| \leq k} \int_{\Omega} | D^{ \alpha}f(x) | ^p dx \right)^{1/p} & {\text{ if }} 1 \leq p < \infty, \cr & \cr \max_{ | \alpha| \leq k } {\text{essup}}_{x \in \Omega} | D^{ \alpha}f(x)| & {\text{ for }} p= \infty.\end{cases} \] Section 2 is devoted to embeddings of Sobolev spaces into the scale of \(L^p( \Omega, E)\)-spaces, where \(E\) is a finite dimensional Hilbert space, and to an analysis of the orthogonal projection (called Sobolev projection) onto the image of the embedding. In particular, one has that, for \( \Omega = {\mathbb R}^n\) and \( 1 < p < \infty\), \(L^p_{(k)}( {\mathbb R}^n)\) is always isomorphic to \(L^p( {\mathbb R}^n)\). This result is extended in section 3 to open subsets \( \Omega \subset {\mathbb R}^n\) which admit a linear extension operator from \(L^p_{(k)}( \Omega)\) into \(L^p_{(k)}({\mathbb R}^n)\). Here one needs Mityagin’s theorem on the isomorphism of spaces of \(k\)-times continuously differentiable functions whose complete proof is also included in section 3.
Section 4 discusses Sobolev spaces for the indices \(p=1\) and \(p= \infty\). Among other results, it is shown that, if \(n=2,3, \dots\) and \(k=1,2, \dots\), then \(L^1_{(k)}( \Omega)\) is not an \({\mathcal L}^1\)-space and \(L^{ \infty}_{(k)}( \Omega)\) is not an quotient of an \( {\mathcal L}_{ \infty}\)-space. Although the spaces of \(k\)-times continuously differentiable functions (for \(k \geq 1\) and \(n \geq 2\)) are not \({\mathcal L}_{ \infty}\)-spaces (a fact also shown in section 4), they share many properties with spaces of continuous functions. This is discussed in section 5.
Section 6 is devoted to Sobolev embedding theorems. Here also Lorentz and Besov spaces are mentioned. Section 7 applies the real and complex interpolation methods to Sobolev spaces. Finally, section 8 reflects, according to the authors, some of their special research interests. Here anisotropic Sobolev spaces \(L_S^p( \Omega)\) are investigated, where \[ L_S^p( \Omega) = \{ f: \Omega \rightarrow {\mathbb K} : D^{ \alpha}f \text{ exists and } D^{ \alpha}f \in L^p( \Omega) \text{ for all } \alpha \in S \} \] and \(S \subset {\mathbb Z}_+^n\) is a finite non-empty set such that \( \beta \in S\) whenever \( \alpha \in S\) and \( \beta \leq \alpha\) (componentwise). The article concludes with an extensive reference list.
For the entire collection see [Zbl 1013.46001].

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46B03 Isomorphic theory (including renorming) of Banach spaces
46-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to functional analysis