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Convolution in the harmonic Hardy class $ h\sp p$ with $ 0<p<1$


Author: Miroslav Pavlović
Journal: Proc. Amer. Math. Soc. 109 (1990), 129-134
MSC: Primary 31A05; Secondary 30D55
DOI: https://doi.org/10.1090/S0002-9939-1990-1012937-7
MathSciNet review: 1012937
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Abstract: It is proved that if $ u \in {h^p},0 < p < 1$, and $ v \in {h^q},q \geq p$, then

$\displaystyle {M_q}(u * v,r) = 0\left( {{{(1 - r)}^{1 - 1/p}}} \right),r \to 1 - ,$

where $ u*v$ stands for the convolution of $ u$ and $ v$.

References [Enhancements On Off] (What's this?)

  • [1] P. L. Duren, Theory of $ {H^p}$ spaces, Academic Press, New York, 1970. MR 0268655 (42:3552)
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  • [3] M. Pavlović, An inequality for the integral means of a Hadamard product, Proc. Amer. Math. Soc. 103 (1988) 404-406. MR 943056 (89h:30002)
  • [4] -, Mean values of harmonic conjugates in the unit disc, Complex Variables 10 (1988), 53-65. MR 946099 (89h:30052)
  • [5] J. H. Shapiro, Linear topological properties of the harmonic Hardy class $ {h^p}$ with $ 0 < p < 1$, Illinois J. Math. 29 (1985), 311-339. MR 784526 (86f:46023)

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DOI: https://doi.org/10.1090/S0002-9939-1990-1012937-7
Article copyright: © Copyright 1990 American Mathematical Society

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