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GENERALIZED HARDY–CESÀRO OPERATORS BETWEEN WEIGHTED SPACES

Part of: Integral, integro-differential, and pseudodifferential operators Special classes of linear operators Integral transforms, operational calculus

Published online by Cambridge University Press:  28 January 2018

THOMAS VILS PEDERSEN Open the ORCID record for THOMAS VILS PEDERSEN [Opens in a new window]
THOMAS VILS PEDERSEN*
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark e-mail: vils@math.ku.dk
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Abstract

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We characterize those non-negative, measurable functions ψ on [0, 1] and positive, continuous functions ω1 and ω2 on ℝ+ for which the generalized Hardy–Cesàro operator

$$\begin{equation*}(U_{\psi}f)(x)=\int_0^1 f(tx)\psi(t)\,dt\end{equation*}$$
defines a bounded operator Uψ: L1(ω1) → L1(ω2) This generalizes a result of Xiao [7] to weighted spaces. Furthermore, we extend Uψ to a bounded operator on M(ω1) with range in L1(ω2) ⊕ ℂδ0, where M(ω1) is the weighted space of locally finite, complex Borel measures on ℝ+. Finally, we show that the zero operator is the only weakly compact generalized Hardy–Cesàro operator from L1(ω1) to L1(ω2).

MSC classification

Primary: 44A15: Special transforms (Legendre, Hilbert, etc.)
Secondary: 47B34: Kernel operators 47B38: Operators on function spaces (general) 47G10: Integral operators

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 61 , Issue 1 , January 2019 , pp. 13 - 24
Copyright
Copyright © Glasgow Mathematical Journal Trust 2018 

References

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