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(ω
1) →
L1(ω
2) This generalizes a result of Xiao [
7] to weighted spaces. Furthermore, we extend
Uψ to a bounded operator on
M(ω
1) with range in
L1(ω
2) ⊕ ℂδ
0, where
M(ω
1) 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(ω
1) to
L1(ω
2).