Function spaces and potential theory.

*(English)*Zbl 0834.46021
Grundlehren der Mathematischen Wissenschaften. 314. Berlin: Springer- Verlag. viii, 366 p. (1995).

Potential theory is one of the outstanding theories in analysis since more than 100 years. It grew out of physics, some of the best mathematicians of our century contributed to it. In our time it became clear that it is closely connected with functional analysis, the theory of distributions, Fourier analysis and (more recent) fractal geometry. Furthermore, the study of nonlinear equations in that context led to the creation of a new, nonlinear potential theory. The two authors contributed substantially to this new theory in the last few years. Their book is devoted to the “interplay of potential theory and function spaces, with the purpose of studying the properties of functions belonging to Sobolev spaces, or to some of their natural extensions, such as Bessel potential spaces, Besov spaces, and Lizorkin-Triebel spaces”. Besides basic facts there is no substantial overlapping with other books on function spaces. The two authors are experts in nonlinear potential theory and function spaces. The book is very well written and will attract a lot of attention (which it highly deserves).

The book has 11 chapters. 1. Preliminaries (basic facts on functional analysis and classical spaces). 2. \(L^2\)-capacities and nonlinear potentials (diverse types of capacities). 3. Estimates for Bessel and Riesz potentials. 4. Besov and Lizorkin-Triebel spaces. 5. Metric properties of capacities. 6. Continuity properties (Lebesgue points, thin sets, etc.). 7. Trace and imbedding theorems (connected with capacities). 8. Poincaré type inequalities. 9. An approximation theorem. 10. Two theorems of Netrusov. 11. Rational and harmonic approximation.

The book has 11 chapters. 1. Preliminaries (basic facts on functional analysis and classical spaces). 2. \(L^2\)-capacities and nonlinear potentials (diverse types of capacities). 3. Estimates for Bessel and Riesz potentials. 4. Besov and Lizorkin-Triebel spaces. 5. Metric properties of capacities. 6. Continuity properties (Lebesgue points, thin sets, etc.). 7. Trace and imbedding theorems (connected with capacities). 8. Poincaré type inequalities. 9. An approximation theorem. 10. Two theorems of Netrusov. 11. Rational and harmonic approximation.

Reviewer: H.Triebel (Jena)

##### MSC:

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

31-02 | Research exposition (monographs, survey articles) pertaining to potential theory |

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

31C45 | Other generalizations (nonlinear potential theory, etc.) |

31B15 | Potentials and capacities, extremal length and related notions in higher dimensions |

31C15 | Potentials and capacities on other spaces |

30E10 | Approximation in the complex plane |