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Embedding derivatives of weighted Hardy spaces into Lebesgue spaces1

Published online by Cambridge University Press:  24 October 2008

Daniel Girela ,
María Lorente  and
María Dolores Sarrión
Daniel Girela
Affiliation:
Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain
María Lorente
Affiliation:
Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain
María Dolores Sarrión
Affiliation:
Departamento de Economía Aplicada (Estad. y Econom.), Facultad de C. Económicas y Empresariales, Universidad de Málaga, 29071 Málaga, Spain
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Abstract

Let 1 ≤ p < ∞ and let ω be a non-negative function defined on the unit circle T which satisfies the Ap condition of Muckenhoupt. The weighted Hardy space Hp(ω) consists of those functions f in the classical Hardy space H1 whose boundary values belong to Lp(ω). Recently McPhail (Studia Math. 96, 1990) has characterized those positive Borel measures μ on the unit disc Δ for which Hp(ω) is continuously contained in Lp(dμ). In this paper we study the question of finding necessary and sufficient conditions on a positive Borel measure μ on Δ for the differentiation operator D defined by Df = f′ to map Hp(ω) continuously into Lp(dμ). We prove that a necessary condition is that there exists a positive constant C such that

where for any interval I ⊂ T,

We prove that this condition is also sufficient in some cases, namely for 2 ≤ p < ∞ and ω(et0) = |θ|α, (|θ| ≤ π), – 1 < α < p – 1, but not in general. In the general case we prove the sufficiency of a condition which is slightly stronger than (A).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 116 , Issue 1 , July 1994 , pp. 151 - 166
Copyright
Copyright © Cambridge Philosophical Society 1994

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References

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