A theorem of Lohwater and Piranian

Author:
Arthur A. Danielyan

Journal:
Proc. Amer. Math. Soc. **144** (2016), 3919-3920

MSC (2010):
Primary 30H05, 30H10

Published electronically:
March 17, 2016

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Abstract | References | Similar Articles | Additional Information

Abstract: By a well-known theorem of Lohwater and Piranian, for any set on of type and of measure zero there exists a bounded analytic function in which fails to have radial limits exactly at the points of . We show that this theorem is an immediate corollary of Fatou's interpolation theorem of 1906.

**[1]**E. F. Collingwood and A. J. Lohwater,*The theory of cluster sets*, Cambridge Tracts in Mathematics and Mathematical Physics, No. 56, Cambridge University Press, Cambridge, 1966. MR**0231999****[2]**Kenneth Hoffman,*Banach spaces of analytic functions*, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962. MR**0133008****[3]**S. V. Kolesnikov,*On the sets of nonexistence of radial limits of bounded analytic functions*, Mat. Sb.**185**(1994), no. 4, 91–100 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math.**81**(1995), no. 2, 477–485. MR**1272188**, 10.1070/SM1995v081n02ABEH003547**[4]**Paul Koosis,*Introduction to 𝐻_{𝑝} spaces*, London Mathematical Society Lecture Note Series, vol. 40, Cambridge University Press, Cambridge-New York, 1980. With an appendix on Wolff’s proof of the corona theorem. MR**565451****[5]**A. J. Lohwater and G. Piranian,*The boundary behavior of functions analytic in a disk*, Ann. Acad. Sci. Fenn. Ser. A. I.**1957**(1957), no. 239, 17. MR**0091342**- [6] G. Piranian, Review of [3], MR1272188 (95g:30043).

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Additional Information

**Arthur A. Danielyan**

Affiliation:
Department of Mathematics and Statistics, University of South Florida, Tampa, Florida 33620

Email:
adaniely@usf.edu

DOI:
https://doi.org/10.1090/proc/13083

Keywords:
Bounded analytic function,
Fatou's theorem,
radial limit,
$F_\sigma$ set

Received by editor(s):
October 4, 2015

Received by editor(s) in revised form:
October 20, 2015, and November 11, 2015

Published electronically:
March 17, 2016

Communicated by:
Svitlana Mayboroda

Article copyright:
© Copyright 2016
American Mathematical Society