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.
Funding Statement
The second author was supported by Grant-in-Aid for Scientific Research (B), Nos. 15H03621 and 20H01815, and, by Grant-in-Aid for Scientific Research (C), No. 21K03304, Japan Society for the Promotion of Science. The third author was supported by Grant-in-Aid for Scientific Research (C), No. 19K03543, Japan Society for the Promotion of Science. The main part of this work was developed when the first author was a graduated student at Ibaraki University.
Acknowledgments
The authors would like to thank the referee for her/his careful reading and many useful comments.
Citation
Kohei Amagai. Eiichi Nakai. Gaku Sadasue. "Generalized Fractional Integral Operators Based on Symmetric Markovian Semigroups with Application to the Heisenberg Group." Taiwanese J. Math. 27 (1) 113 - 139, February, 2023. https://doi.org/10.11650/tjm/220904
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