CONICET | Buscador de Institutos y Recursos Humanos

We characterize the pairs of weights $(v,w)$ for which theoperator $Tf(x)=g(x)int_{s(x)}^{h(x)}f$ with $s$ and $h$increasing and continuous functions is of strong type$(p,q)$ or weak type $(p,q)$ with respect to the pair$(v,w)$ in the case $0<q<p$ and $1<p<infty$. The resultfor the weak type is new while the characterizations forthe strong type improve the ones given by H. P. Heinig andG. Sinnamon. In particular, we do not assumedifferentiability properties on $s$ and $h$ and we obtainthat the strong type inequality $(p,q)$, $q<p$, ischaracterized by the fact that the function $Phi(x)=sup left(int_c^dg^qw ight)^{1/p} left(int_{s(d)}^{h(c)}v^{1-p^prime} ight)^{1/p^prime}$ belongs to $L^{r}(g^qw)$, where $1/r=1/q-1/p$ and the supremum is taken over all $c$ and $d$such that $c leq x leq d$ and $s(d) leq h(c)$.