Harmonic measure

A concept in the theory of harmonic functions (cf. Harmonic function) connected with estimating the modulus of an analytic function inside a domain if certain bounds on the modulus on the boundary of the domain are known , . Let be a bounded open set in the Euclidean space , ; let be the boundary of ; and let be a finite real-valued continuous function on . To each such function there corresponds a unique harmonic function on which, for , is a generalized solution of the Dirichlet problem. If the point is assumed to be fixed, the functional will define on the compact set a positive Radon measure , which is called the harmonic measure at the point . The formula for the representation of the generalized solution of the Dirichlet problem, obtained by Ch.J. de la Vallée-Poussin by the balayage method, is valid for all functions which are continuous on . Moreover, if is an arbitrary Borel set on , the harmonic measure , , of at is equal to the value at of the generalized solution of the Dirichlet problem for the characteristic function , , of .

The basic properties of a harmonic measure are: is a harmonic function of the point in ;  if is a domain and even at a single point , then .

In the last-named case is known as a set of harmonic measure zero. If a compact set has harmonic measure zero with respect to some domain , , i.e. , then it has harmonic measure zero with respect to all other domains, i.e. is a set of absolute harmonic measure zero. A set has absolute harmonic measure zero if and only if it has zero (harmonic) capacity.

As regards applications to the theory of functions of a complex variable, the dependence of a harmonic measure on the domain is of special importance. This dependence is expressed by the principle of harmonic measure (cf. Harmonic measure, principle of), which states that a harmonic measure does not decrease under mappings of the domain realized by univalent analytic functions , . In particular, a harmonic measure remains unchanged under a one-to-one conformal mapping.

Explicit computations of harmonic measures are possible only for the simplest domains (mainly for the disc, for the sphere, for a half-plane, and for a half-space; see Poisson integral). This is the reason for the importance of the various estimation methods for harmonic measure , , , , which are mainly based on the principle of extension of domain (cf. Extension of domain, principle of). In the simplest form, for , this principle consists in the following: Let a finitely-connected domain be bounded by a finite number of Jordan curves and let be an arc on . Then, if the domain is extended in some way across the complementary part of the boundary, the harmonic measure can only increase.