Regularity of the generalized Poisson operator
J. Korean Math. Soc. 2022 Vol. 59, No. 1, 129-150
Published online November 25, 2021
Printed January 1, 2022
Pengtao Li, Zhiyong Wang, Kai Zhao
Qingdao University; Qingdao University; Qingdao University
Abstract : Let $L=-\Delta+V$ be a Schr\"{o}dinger operator, where the potential $V$ belongs to the reverse H\"{o}lder class. In this paper, by the subordinative formula, we investigate the generalized Poisson operator $P^{L}_{t,\sigma}$, $0<\sigma<1$, associated with $L$. We estimate the gradient and the time-fractional derivatives of the kernel of $P^{L}_{t,\sigma}$, respectively. As an application, we establish a Carleson measure characterization of the Campanato type space $\mathcal{C}^{\gamma}_{L}(\mathbb{R}^{n})$ via $P^{L}_{t,\sigma}$.
Keywords : Regularities, Schr\"{o}dinger operators, the generalized Poisson operators, Campanato type spaces
MSC numbers : Primary 42B35, 35J10, 42B38
Supported by : This work was financially supported by the National Natural Science Foundation of China (No. 12071272) and Shandong Natural Science Foundation of China (Nos. ZR2020MA004, ZR2017JL008).
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