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

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
How to Cite This Entry:
Harmonic measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Harmonic_measure&oldid=47182
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article