# Journal of theKorean Mathematical SocietyJKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

## Article

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J. Korean Math. Soc. 2022; 59(1): 129-150

Published online January 1, 2022 https://doi.org/10.4134/JKMS.j210224

## Regularity of the generalized Poisson operator

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