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- Title
Some properties of Van Koch’s curves.
- Authors
Ponomarev, Stanislav
- Abstract
We investigate the properties of an integral operator T with a Cauchy kernel. The operator acts from L ∞(Γ, μ), where Γ is a Van Koch curve, to the space of functions ℂ → ℂ. We prove that the range of T is nontrivial and lies in the space AC(Γ) of functions continuous in ℂ, vanishing at ∞, and analytic outside Γ. We also show that T is injective and compact while satisfying some special functional equation. These results may be regarded as a natural continuation of our research on the problem of AC-removability of quasiconformal curves whose solution was announced in [1] for the first time and supplemented later with some other properties of Van Koch’s curves [2, 3]. In this paper the problem is discussed in a more general setting and, in particular, all important details lacking in [1] are given. Some open problems are formulated.
- Subjects
CAUCHY integrals; INTEGRAL operators; QUASICONFORMAL mappings; COMPACT operators; MATHEMATICAL research
- Publication
Siberian Mathematical Journal, 2007, Vol 48, Issue 6, p1046
- ISSN
0037-4466
- Publication type
Article
- DOI
10.1007/s11202-007-0107-0