# Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator Academic Article

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### abstract

• 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.

### publication date

• January 1, 1984