Weighted generalized weak type inequalities for modified Hardy operators (Q1585380)

We consider the operator \(T_gf(x)= g(x)\int^x_0 f\), where \(g\) is a positive nonincreasing function, and characterize the pairs of positive measurable functions \((u,v)\) such that the generalized weak-type inequality \[ \Phi^{-1}_2\Biggl( \Phi_2(\lambda) \int_{\{x\in (0,\infty);|T_gf(x)|> \lambda\}}u\Biggr)\leq \Phi^{-1}_1 \Biggl(\int^\infty_0 \Phi_1(K|f|) v\Biggr) \] holds, where either \(\Phi_1\) is an \(N\)-function and \(\Phi_2\) is a positive increasing function such that \(\Phi_1\circ \Phi^{-1}_2\) is countably subadditive or \(\Phi_1(t)= t\) and \(\Phi_2\) is a positive increasing function whose inverse is countably subadditive. These results generalize the weak-type \((p,q)\) inequality for \(T_g\) in the case \(1\leq p\leq q\). The weighted generalized weak-type inequality considered in the paper had been studied previously by Q. Lai in the case \(g= 1\) and by S. Bloom and R. Kerman for nondecreasing \(g\).