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.
|||T. Carleman, "Sur les fonctions inverses des fonctions entières d'ordre fini" Ark. Mat. , 15 : 10 (1921) pp. 1–7|
|||F. Nevanlinna, R. Nevanlinna, "Ueber die Eigenschaften einer analytischen Funktion in der Umgebung einer singulären Stelle oder Linie" Acta Soc. Sci. Fennica , 50 : 5 (1922) pp. 1–46|
|||Ch.J. de la Vallée-Poussin, Ann. Inst. H. Poincaré , 2 (1932) pp. 169–232|
|||R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German)|
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|||M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959)|
|||K. Haliste, "Estimates of harmonic measure" Ark. Mat. , 6 : 1 (1965) pp. 1–31|
For domains in very precise estimates for harmonic measure in terms of Hausdorff measure have been obtained recently. Let be a continuous increasing function for , , and let be a Borel set. Let denote the Hausdorff measure with respect to of . Let be a domain in and set . Makarov's theorems [a3] are: 1) Suppose that is simply connected. If , then is singular with respect to , i.e. . 2) There exist constants , with the following property: Let
Then for every Jordan domain , is absolutely continuous with respect to , i.e. . However, there exists a Jordan domain with .
Next (B. Øksendal, Jones, Wolff): If , then for every domain in , .
|[a1]||C. Constantinescu, A. Cornea, "Potential theory on harmonic spaces" , Springer (1972)|
|[a2]||J.B. Garnett, "Applications of harmonic measure" , Wiley (Interscience) (1986)|
|[a3]||N. Makarov, "On the distortion of boundary sets under conformal mappings" Proc. London Math. Soc. , 51 (1985) pp. 369–384|
Harmonic measure. E.D. Solomentsev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Harmonic_measure&oldid=11907