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On the radial projection of harmonic measure. (Russian) Zbl 0791.31003
Marchenko, V. A. (ed.), Operator theory, subharmonic functions. Collection of scientific works. Kiev: Naukova Dumka. 95-102 (1991).
Let $$D$$ denote a domain in $$\mathbb{C}$$ containing the origin, $$E$$ a part of the boundary of $$D$$, $$\widehat{E}$$ the radial projection of $$E$$ on the unit circle, $$\widehat{\omega}_ D(\widehat{E}) = \omega(0,E,D)$$, the harmonic measure of $$E$$ with respect to $$D$$. Let $$m$$ denote a probability measure on the unit circle.
The question studied here is whether there exists a domain $$D$$ such that $$m(\widehat{E}) = \omega(0,E,D)$$. This is solved with NASC in the class of star-like domains $$D$$. Applications are given to the growth of subharmonic functions [cf. the author, Sov. Math., Dokl. 41, No. 3, 460- 462 (1990); translation from Dokl. Akad. Nauk SSSR 312, No. 3, 536-538 (1990; Zbl 0722.31001)].
For the entire collection see [Zbl 0752.00029].

##### MSC:
 31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions 31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
##### Keywords:
star-like domain; subharmonic function