A note on norm inequalities for integral operators on cones

Abstract:

Norm inequalities for the Riemann-Liouville operator ${R_r}f(x) = \int _{\langle 0,x\rangle } {\Delta _V^{r - 1}(x - t)f(t)dt}$ and Weyl operator ${W_r}f(x) = \int _{\langle x,\infty \rangle } {\Delta _V^{r - 1}(t - x)f(t)dt}$ on cones in ${R^d}$ have been obtained in the case $r \geqslant 1$ [7]. In this note, these inequalities are further extended to the case $r < 1$. The question of whether the Hardy operator $Hf(x) = \int _{\langle 0,x\rangle } {f(t)dt}$ on cones is bounded from ${L^p}(\Delta _V^\alpha (X))$ to ${L^q}(\Delta _V^\beta (x))\;(q < p)$ is also solved.