On the boundary behavior of conjugate harmonic functions
It is proved that if a harmonic function u on the unit disk D in C has angular limits on a measurable set E of the unit circle, then its conjugate harmonic function v in D also has (finite !) angular limits a.e. on E and both boundary functions are measurable on E. The result is extended to arbitrar...
Saved in:
| Date: | 2017 |
|---|---|
| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Інститут прикладної математики і механіки НАН України
2017
|
| Series: | Праці Інституту прикладної математики і механіки НАН України |
| Online Access: | http://dspace.nbuv.gov.ua/handle/123456789/145115 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | On the boundary behavior of conjugate harmonic functions / V.I. Ryazanov // Праці Інституту прикладної математики і механіки НАН України. — Слов’янськ: ІПММ НАН України, 2017. — Т. 31. — С. 117-123. — Бібліогр.: 19 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| Summary: | It is proved that if a harmonic function u on the unit disk D in C has angular limits on a measurable set E of the unit circle, then its conjugate harmonic function v in D also has (finite !) angular limits a.e. on E and both boundary functions are measurable on E. The result is extended to arbitrary Jordan domains with rectifiable boundaries in terms of angular limits and of the natural parameter. This result is essentially based on the Fatou theorem on angular limits of bounded analytic functions and on the construction of Luzin and Priwalow to their uniqueness theorem for analytic and meromorphic functions. The result will have interesting applications to the study of the various Stieltjes integrals in the theory of harmonic and analytic functions and, in particular, of the Hilbert–Stieltjes inyegral. |
|---|