Geodesic
GENERALIZED HARDY–CESÀRO OPERATORS BETWEEN WEIGHTED SPACES
Glasgow mathematical journal, Tome 61 (2019) no. 1, pp. 13-24

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DOI

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). 
PEDERSEN, THOMAS VILS. GENERALIZED HARDY–CESÀRO OPERATORS BETWEEN WEIGHTED SPACES. Glasgow mathematical journal, Tome 61 (2019) no. 1, pp. 13-24. doi: 10.1017/S0017089517000398
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     author = {PEDERSEN, THOMAS VILS},
     title = {GENERALIZED {HARDY{\textendash}CES\`ARO} {OPERATORS} {BETWEEN} {WEIGHTED} {SPACES}},
     journal = {Glasgow mathematical journal},
     pages = {13--24},
     year = {2019},
     volume = {61},
     number = {1},
     doi = {10.1017/S0017089517000398},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089517000398/}
}
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