Three-weight generalization of Hardy's inequality (Q1337857)

Let \(-\infty\leq a< b\leq \infty\), \(r\), \(\rho\) and \(v\) be nonnegative measurable functions on \((a, b)\). Let \[ Kf(x)= \int^x_a k(x, t) f(t)\,dt. \] Here, the nonnegative kernel \(k\) satisfies the condition \(k(x, s)\leq dk(x, t)\) for \(a< t\leq s\leq x< b\), where the constant \(d\geq 1\) is independent of \(t\), \(s\) and \(x\). Moreover, let \(Pf(x)= \int^x_a f(t)\,dt\). For the case in which \(1< p\leq q< \infty\) necessary and sufficient conditions on the functions \(r\), \(\rho\), and \(v\) are found for the following inequality to hold \[ |rPf|_q\leq C(|\rho f|_p+ |vK|f||_p), \] where \(|\cdot|_s\) is the norm in \(L_s(a, b)\) and the constant \(C\) is independent of \(f\).