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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).