When solving the Laplace equation in a domain, we can look for a harmonic function with prescribed continuous boundary data (at least in a generalised sense). This is the Dirichlet problem. Through the Riesz representation theorem, we can identify the functional that evaluates the solution at a given point with the so-called harmonic measure. This construction works in an arbitrary space dimension. However, the planar case differs in many aspects and leads to a separate rapidly developing theory. This volume builds the planar theory with the aim of reaching several distinguished results with an emphasis on recent achievements. The first four chapters contain the foundations of the subject; the themes include conformal mappings, hyperbolic metrics, potential theory and extremal distance. Chapter V presents Teichmüller's Modulsatz and some results on angular derivatives. Chapters VI and VIII are devoted to the relation between the harmonic measure and the Hausdorff measure (Makarov's theorem) and to Brennan's conjecture about Sobolev regularity of conformal mappings. The topics of chapter VII are conformal images of the unit disc (the relation of their geometry to the properties of the conformal mapping), Bloch functions, Muckenhoupt's weights and quasi-circles. Chapter VIII studies domains with an infinite number of components. In chapter IX, a discussion involving the Lusin area function, the Schwarzian derivative and the Jones square is applied to obtain deep results on univalent sums. A nontrivial amount of material, which is regarded as preliminary with respect to the topic of the book, is summarised in the appendices. Each chapter is equipped with numerous exercises. The book can be warmly recommended to students and researchers with a deep interest in analysis. It is an excellent preparation for serious work in complex analysis or potential theory.

Reviewer:

jama