Average radial limits in weighted Hardy spaces

Authors:
Alec Matheson and David C. Ullrich

Journal:
Proc. Amer. Math. Soc. **97** (1986), 691-694

MSC:
Primary 30D40; Secondary 30D50

DOI:
https://doi.org/10.1090/S0002-9939-1986-0845989-4

MathSciNet review:
845989

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Abstract: Weighted Hardy spaces are defined in the unit disc by specifying the rate of growth of th means near the boundary. Although a function in one of these spaces need have no radial limits, it is shown that in certain of these spaces "average radial limits" exist over an interval on the boundary. An integral representation in terms of these average radial limits is given, with an application to the question of existence of (pointwise) radial limits.

**[D]**P. L. Duren,*Theory of**spaces*, Academic Press, New York, 1970. MR**0268655 (42:3552)****[HL]**G. H. Hardy and J. E. Littlewood,*Theorems concerning mean values of analytic or harmonic functions*, Quart. J. Math. Oxford Ser.**8**(1937), 161-171. MR**0006581 (4:8d)****[L]**L. H. Loomis,*The converse of the Fatou theorem for positive harmonic functions*, Trans. Amer. Math. Soc.**53**(1943), 239. MR**0007832 (4:199d)****[M]**A. Matheson,*A multiplier theorem for analytic functions of slow mean growth*, Proc. Amer. Math. Soc.**77**(1979), 53-57. MR**539630 (80j:30048)****[U]**D. C. Ullrich,*Radial limits of Bloch functions in the unit disc*, Bull. London Math. Soc.**18**(1986). MR**838805 (87k:30059)**

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DOI:
https://doi.org/10.1090/S0002-9939-1986-0845989-4

Article copyright:
© Copyright 1986
American Mathematical Society