Generalized Fractional Integral Operators Based on Symmetric Markovian Semigroups with Application to the Heisenberg Group

Abstract

It is known that the fractional integral operator $\mathcal{I}_{\alpha}$ based on a symmetric Markovian semigroup with Varopoulos dimension $d$ is bounded from $L^p$ to $L^q$, if $0 \lt \alpha \lt d$, $1 \lt p \lt q \lt \infty$ and $-d/p + \alpha = -d/q$, like the usual fractional integral operator defined on the $d$ dimensional Euclidean space. We introduce generalized fractional integral operators based on symmetric Markovian semigroups and extend the $L^p$-$L^q$ boundedness to Orlicz spaces. We also apply the result to the semigroup associated with the diffusion process generated by the sub-Laplacian on the Heisenberg group. Moreover, we show necessary and sufficient conditions for the boundedness of the generalized fractional integral operator on the space of homogeneous type and apply them to the Heisenberg group.