##
**Weighted functional spaces and the spectrum of differential operators.
(Vesovye funktsional’nye prostranstva i spektr differentsial’nykh operatorov.)**
*(Russian.
English summary)*
Zbl 0651.46037

Moskva: Nauka. 286 p. R. 3.8 (1988).

The book deals with weighted function spaces in one and several dimensions, related embedding theorems, compactness of the embeddings, sharp estimates of approximation numbers and applications to the spectral theory of ordinary and partial differential operators.

The book has 9 chapters and an introduction. Introduction: \(L_ p\) spaces, Sobolev spaces, capacity. I (General theorems of compactness in spaces of smooth functions): Semi-group approach to (unweighted) function spaces of Sobolev-Besov type. II (On the embedding of some spaces into the space of continuous functions): Weighted Sobolev spaces, continuous embedding in spaces of continuous functions, density of \(C_ 0^{\infty}\). III (Embedding theorems and compactness of weighted spaces of Sobolev type. One-dimensional case): Detailed study of the embedding \(W\subset L\), where W and L are normed by \[ \| f\|_ W=[\int_{\Omega}(| f'(t)|^ p+| \nu (t)f(t)|^ p)dt]^{1/p} \] and \[ \| f\|_ L=(\int_{\Omega}| r(t)f(t)|^ qdt)^{1/q}, \] respectively, new sharp results, measure of non-compactness, compactness of embeddings. IV (Estimates for approximation numbers for embeddings of weighted Sobolev spaces. One- dimensional case): Approximation numbers for \(\ell_ p\) spaces, for embeddings in the sense of the preceding chapter. V (Estimates for approximation numbers. Several dimensions): Beside an obvious extension of L, the above space W is now normed by \[ \| f\|_ W=[\int_{\Omega}(\sum_{| \alpha | =\ell}| \mu (x)D^{\alpha}f(x)|^ p+| \nu (x)f(x)|^ p)dx]^{1/p}, \] n-dimensional counterpart of Chapter 3, capacity, approximation numbers. VI (Application of embedding theorems and approximation to study the spectrum of semi-bounded operators): Bilinear forms generating differential operators of Schrödinger type. Spectral assertions. VII (On the smoothness of solutions of Sturm-Liouville equation): One- dimensional and n-dimensional equations, spectral assertions. VIII (Density of finite functions): Density of \(C_ 0^{\infty}\) in weighted Sobolev spaces. IX (Difference embedding theorems): Some assertions of the above type in sequence and matrix spaces.

The text is based on the research of the authors. A working knowledge of basic assertions in functional analysis is sufficient to understand the text.

The book has 9 chapters and an introduction. Introduction: \(L_ p\) spaces, Sobolev spaces, capacity. I (General theorems of compactness in spaces of smooth functions): Semi-group approach to (unweighted) function spaces of Sobolev-Besov type. II (On the embedding of some spaces into the space of continuous functions): Weighted Sobolev spaces, continuous embedding in spaces of continuous functions, density of \(C_ 0^{\infty}\). III (Embedding theorems and compactness of weighted spaces of Sobolev type. One-dimensional case): Detailed study of the embedding \(W\subset L\), where W and L are normed by \[ \| f\|_ W=[\int_{\Omega}(| f'(t)|^ p+| \nu (t)f(t)|^ p)dt]^{1/p} \] and \[ \| f\|_ L=(\int_{\Omega}| r(t)f(t)|^ qdt)^{1/q}, \] respectively, new sharp results, measure of non-compactness, compactness of embeddings. IV (Estimates for approximation numbers for embeddings of weighted Sobolev spaces. One- dimensional case): Approximation numbers for \(\ell_ p\) spaces, for embeddings in the sense of the preceding chapter. V (Estimates for approximation numbers. Several dimensions): Beside an obvious extension of L, the above space W is now normed by \[ \| f\|_ W=[\int_{\Omega}(\sum_{| \alpha | =\ell}| \mu (x)D^{\alpha}f(x)|^ p+| \nu (x)f(x)|^ p)dx]^{1/p}, \] n-dimensional counterpart of Chapter 3, capacity, approximation numbers. VI (Application of embedding theorems and approximation to study the spectrum of semi-bounded operators): Bilinear forms generating differential operators of Schrödinger type. Spectral assertions. VII (On the smoothness of solutions of Sturm-Liouville equation): One- dimensional and n-dimensional equations, spectral assertions. VIII (Density of finite functions): Density of \(C_ 0^{\infty}\) in weighted Sobolev spaces. IX (Difference embedding theorems): Some assertions of the above type in sequence and matrix spaces.

The text is based on the research of the authors. A working knowledge of basic assertions in functional analysis is sufficient to understand the text.

Reviewer: H.Triebel

### MSC:

46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |

46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

47B10 | Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) |

35S05 | Pseudodifferential operators as generalizations of partial differential operators |