Potential theory - selected topics
This book is in two parts, separately authored and indexed, and with separate bibliographies. Part II, by \textit{H. Aikawa}, is a unified account of some recent developments in potential theory due mainly to the author. Part I, by \textit{M. Essén}, presents the necessary potential theoretic background in an informal style, with questions to think about, side remarks and helpful diagrams. The reader is assumed to be familiar with integration and distribution theory, and some functional analysis. The approach of Part I is similar in spirit to the texts of \textit{L. Carleson} [Selected problems on exceptional sets, Van Nostrand, Princeton (1967; Zbl 0189.10903); reprint Wadsworth (1983; Zbl 0505.00034)] and \textit{N. S. Landkof} [Foundations of modern potential theory, Springer, Berlin (1972; Zbl 0253.31001)]. It does not attempt to be a comprehensive treatise on potential theory, but does cover, in some form, a substantial part of the core of the subject. Where proofs are omitted, a convenient reference is supplied. The account begins with a discussion of potentials and capacity arising from a certain class of kernels. The text presents several equivalent formulations of capacity, the capacitability of analytic sets, the relationship between Hausdorff measures and capacities, and Cantor sets which illustrate the subtleties of this relationship. Attention is then directed to potentials arising from the M. Riesz kernel \(k_\alpha(x)= |x|^{\alpha-N}\), where \(0< \alpha < N\). There is an account of \(\alpha\)-energy, \(\alpha\)-capacity and associated equilibrium measures. For the potential theory of the Laplace operator, there is a discussion of Green potentials and reduced functions, the fine topology and thin sets. Then, in the context of a ball or half-space, minimally thin and rarefied sets are introduced and characterized by Wiener-type criteria involving Green energy and Green mass, respectively. Singular integral methods are used to establish the Beurling-Mazya-Dahlberg result that, if \(E\) is minimally thin at infinity for the half-space \(D=(0,\infty) \times\mathbb{R}^{N-1}\), then \(\int_E(1+ |x|)^{-N} dx<\infty\). Finally, the author presents some of his own results on Wiener-type criteria for minimally thin and rarefied sets, this time in terms of ordinary capacity and a Whitney cube decomposition \(\{Q_k\}\) of \(D\). This is a natural point of transition from Part I to Part II, for the following reason. Green energy \(\gamma\) for \(D\) is known to be subadditive; that is, if \(E= \cup_kE_k\), then \(\gamma(E) \leq\sum_k \gamma(E_k)\). In the course of proving the above Wiener-type criteria, it is shown that, if \(E\subseteq \{x\in D: 2^{-1} \leq|x|\leq 4\}\), then there is an absolute constant \(C\) such that \(\sum\gamma (E \cap Q_k) \leq C\gamma(E)\). Thus Green energy is almost additive for Whitney-type decompositions. Aikawa refers to this property as ``quasi-additivity'', and has established that it holds for several types of capacity. Part II begins with a general account of \(L^p\)-potential theory on \(\mathbb{R}^n\), as developed by N. G. Meyers. There follow estimates of the capacity of balls for kernels of convolution type, comparisons of \(L^p\)-capacity and Hausdorff measure, and estimates concerning the effect of Lipschitz mappings on capacity. Next comes an elementary proof of the capacity strong-type inequality via the weak maximum principle for nonlinear potentials. Quasi-additivity of Riesz capacity is established for a Whitney cube decomposition of \(\mathbb{R}^N \backslash F\), where \(F\) is a nowhere dense closed set of restricted ``dimension''. Quasi-additivity is also discussed for Green energy, which is viewed as the capacity associated with the \(\Theta\)-kernel introduced by \textit{L. Naïm} in her foundational work on minimal thinness. This discussion takes place in the context of ``uniformly \(\Delta\)-regular'' domains, which include Lipschitz and non-tangentially accessible domains. Next is an exploration of the boundary behaviour of harmonic and superharmonic functions on \(D\) via two notions of thinness of a set ``at the boundary'' (rather than at a specified boundary point). This approach leads to generalizations of the Fatou-Naïm-Doob minimal fine limit theorem for \(D\), and of work of Nagel and Stein on the boundary behaviour of harmonic functions. The final section deals with integrability of superharmonic and subharmonic functions on Lipschitz domains \(\Omega\), including sharp bounds (in terms of the Lipschitz constant) on the values of \(p\) for which a positive superharmonic function is \(L^p\)-integrable on \(\Omega\). Appendices cover Choquet's capacitability theorem and the Fatou-Naïm-Doob theorem. Researchers who have an interest in potential theory will find this book a valuable source of background information and a convenient route to some recent deep and significant results in the subject.
potential theory of the Laplace operator
boundary behaviour of harmonic and superharmonic functions
Fatou-Naïm-Doob minimal fine limit theorem
integrability of superharmonic and subharmonic functions
scientific article; zbMATH DE number 945655