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 $ D $ be a bounded open set in the Euclidean space $ \mathbf R ^ {n} $, $ n \geq 2 $; let $ \Gamma = \partial D $ be the boundary of $ D $; and let $ f $ be a finite real-valued continuous function on $ \Gamma $. To each such function $ f $ there corresponds a unique harmonic function $ H _ {f} ( x) $ on $ D $ which, for $ f $, is a generalized solution of the Dirichlet problem. If the point $ x \in D $ is assumed to be fixed, the functional $ H _ {f} ( x) $ will define on the compact set $ \Gamma $ a positive Radon measure $ \omega ( x) = \omega ( x, D) $, which is called the harmonic measure at the point $ x $. The formula for the representation of the generalized solution of the Dirichlet problem,

$$ H _ {f} ( x) = \ \int\limits f ( y) d \omega ( x; D), $$

obtained by Ch.J. de la Vallée-Poussin by the balayage method, is valid for all functions $ f $ which are continuous on $ \Gamma $. Moreover, if $ E $ is an arbitrary Borel set on $ \Gamma $, the harmonic measure $ \omega ( x; E, D) $, $ x \in D $, of $ E $ at $ x $ is equal to the value at $ x $ of the generalized solution of the Dirichlet problem for the characteristic function $ \chi _ {E} ( y) $, $ y \in \Gamma $, of $ E $.

The basic properties of a harmonic measure are: $ \omega ( x; E, D) $ is a harmonic function of the point $ x $ in $ D $;

$$ 0 \leq \omega ( x; E, D) \leq 1; $$

$$ 1 - \omega ( x; E, D) = \omega ( x; \Gamma \setminus E, D); $$

if $ D $ is a domain and $ \omega ( x; E, D) = 0 $ even at a single point $ x \in D $, then $ \omega ( x; E, D) \equiv 0 $.

In the last-named case $ E $ is known as a set of harmonic measure zero. If a compact set $ K \subset \mathbf R ^ {n} $ has harmonic measure zero with respect to some domain $ D $, $ K \subset D $, i.e. $ \omega ( x; K, D \setminus K) = 0 $, then it has harmonic measure zero with respect to all other domains, i.e. $ K $ is a set of absolute harmonic measure zero. A set $ K $ 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 $ D $ 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 $ D $ realized by univalent analytic functions $ w = w( z) $, $ z \in D $. 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 $ D $( 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 $ n = 2 $, this principle consists in the following: Let a finitely-connected domain $ D $ be bounded by a finite number of Jordan curves $ \Gamma $ and let $ \alpha $ be an arc on $ \Gamma $. Then, if the domain $ D $ is extended in some way across the complementary part $ \Gamma \setminus \alpha $ of the boundary, the harmonic measure $ \omega ( z; \alpha , D) $ 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 $ \mathbf C $ very precise estimates for harmonic measure in terms of Hausdorff measure have been obtained recently. Let $ h( t) $ be a continuous increasing function for $ t \geq 0 $, $ h( 0)= 0 $, and let $ E $ be a Borel set. Let $ \Lambda _ {h} ( E) $ denote the Hausdorff measure with respect to $ h $ of $ E $. Let $ D $ be a domain in $ \mathbf C $ and set $ \omega ( E) = \omega ( x; E, D) $. Makarov's theorems [a3] are: 1) Suppose that $ D $ is simply connected. If $ \lim\limits _ {t \rightarrow \infty } h( t) /t = 0 $, then $ \omega $ is singular with respect to $ \Lambda _ {h} $, i.e. $ \omega \perp \Lambda _ {h} $. 2) There exist constants $ C _ {1} $, $ C _ {2} $ with the following property: Let

$$ h _ {i} ( t) = t \mathop{\rm exp} \left \{ C _ {i} \sqrt {\left ( \mathop{\rm log} \frac{1}{t} \right ) \mathop{\rm log} \mathop{\rm log} \mathop{\rm log} \frac{1}{t} } \right \} ,\ i = 1, 2. $$

Then for every Jordan domain $ D $, $ \omega $ is absolutely continuous with respect to $ \Lambda _ {h _ {1} } $, i.e. $ \omega \ll \Lambda _ {h _ {1} } $. However, there exists a Jordan domain $ D $ with $ \omega \perp \Lambda _ {h _ {2} } $.

Next (B. Øksendal, Jones, Wolff): If $ 1 < a \leq 2 $, then for every domain $ D $ in $ \mathbf C $, $ \omega \perp \Lambda _ {( t ^ \alpha ) } $.

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