Weighted norm inequalities for certain integral operators. II

Abstract:

Conditions on nonnegative weight functions $u(x)$ and $\upsilon (x)$ are given which ensure that an inequality of the form ${(\smallint {\left | {Tf(x)} \right |^q}u(x)\;dx)^{1/q}} \leqslant C{(\smallint {\left | {f(x)} \right |^p}\upsilon (x)\;dx)^{1/p}}$ holds for $1 \leqslant q < p < \infty$, where $T$ is an integral operator of the form $\int _{ - \infty }^x {K(x,y)f(y)dy}$ or $\int _x^\infty {K(y,x)f(y)\;dy}$ and $C$ a constant independent of $f$. Specifically a number of inequalities for well-known classical operators are obtained. Inequalities of the above form for $1 \leqslant p \leqslant q < \infty$ were obtained in [1].