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 [1], [2]. 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 [4], [5], [6], [7], 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.

References

[1]	T. Carleman, "Sur les fonctions inverses des fonctions entières d'ordre fini" Ark. Mat. , 15 : 10 (1921) pp. 1–7
[2]	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
[3]	Ch.J. de la Vallée-Poussin, Ann. Inst. H. Poincaré , 2 (1932) pp. 169–232
[4]	R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German)
[5]	G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[6]	M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959)
[7]	K. Haliste, "Estimates of harmonic measure" Ark. Mat. , 6 : 1 (1965) pp. 1–31

Comments

Harmonic measure is an important tool in axiomatic potential theory (cf. Potential theory, abstract), see [a1].

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 , .

References

[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