Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator

Characterizations are obtained for those pairs of weight functions $w,\upsilon$ for which the Hardy operator $Tf(x)={\int }_0^{x}f(s)ds$ is bounded from the Lorentz space $L^{r,s}((0,\mathrm{\infty }),\upsilon dx)$ to $L^{p,q}((0,\mathrm{\infty }),wdx),0>p,q,r,s⩽\mathrm{\infty }$ . The modified Hardy operators $T_{\eta }f(x)=x^{-\eta }Tf(x)$ for $\eta$ real are also treated.