The authors consider the integral operator given by \(Tf(x) = \int^ \infty_ 0 k(x,t) f(t)dt\), where \(k:{\mathcal M} \times \mathbb{R}^ + \to \mathbb{R}^ +\) and \(({\mathcal M},\mu)\) is some measure space. They determine the mapping properties of \(T\) mapping functions in \(L^ p_{dec}\), \(0<p \leq 1\), into functions in \(L^ q_ w\), \(q \geq p\); here \(L^ p_{dec}\) is the class of nonincreasing functions in \(L^ p\), and \(L^ q(w)\) is a weighted \(L^ q\) space. They also prove related weak-type inequalities and apply their results to generalized Hardy operators.