Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator Journal Articles

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