Weighted Norm Inequalities for Certain Integral Operators

Source abstract

Conditions on the nonnegative weight functions $u(x)$ and $v(x)$ are given which ensure that an inequality of the form $(\int {| {(Tf)(x)u(x)} |^q dx} )^{{1 /q}} \leq C(\int {| {f(x)v(x)} |^p dx} )^{{1 /p}}$ holds where T is an integral operator of the form $\int_{ - \infty }^x K (x,y)f(y)dy$ or $\int_x^\infty K (x,y)f(y)dy$ and C is a constant depending on $K,p,q$ but independent of f; the inequality being reversed in case $p,q < 1$. In particular, new inequalities and a unified treatment of several known inequalities are obtained for a class of convolution operators, various fractional integrals and the Laplace transform.