Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator

Abstract:

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 ${L^{r,s}}((0,\infty ),\upsilon dx)$ to ${L^{p,q}}((0,\infty ),w dx),0 < p,q,r,s \leqslant \infty$. The modified Hardy operators ${T_\eta }f(x) = {x^{ - \eta }}Tf(x)$ for $\eta$ real are also treated.