Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2022; 59(6): 1471-1493

Online first article November 10, 2022      Printed November 30, 2022

https://doi.org/10.4134/BKMS.b210839

Copyright © The Korean Mathematical Society.

Estimate for bilinear Calder\'{o}n-Zygmund operator and its commutator on product of variable exponent spaces

Guanghui Lu, Shuangping Tao

Northwest Normal University; Northwest Normal University

Abstract

The goal of this paper is to establish the boundedness of bilinear Calder\'{o}n-Zygmund operator $BT$ and its commutator $[b_{1},b_{2},BT]$ which is generated by $b_{1}, b_{2}\in\mathrm{BMO}(\mathbb{R}^{n})$ (or $\dot{\Lambda}_{\alpha}(\mathbb{R}^{n})$) and the $BT$ on generalized variable exponent Morrey spaces $\mathcal{L}^{p(\cdot),\varphi}(\mathbb{R}^{n})$. Under assumption that the functions $\varphi_{1}$ and $\varphi_{2}$ satisfy certain conditions, the authors proved that the $BT$ is bounded from product of spaces $\mathcal{L}^{p_{1}(\cdot),\varphi_{1}}(\mathbb{R}^{n}) \times\mathcal{L}^{p_{2}(\cdot),\varphi_{2}}(\mathbb{R}^{n})$ into space $\mathcal{L}^{p(\cdot),\varphi}(\mathbb{R}^{n})$. Furthermore, the boundedness of commutator $[b_{1},b_{2},BT]$ on spaces $L^{p(\cdot)}(\mathbb{R}^{n})$ and on spaces $\mathcal{L}^{p(\cdot),\varphi}(\mathbb{R}^{n})$ is also established.

Keywords: Bilinear Calder\'{o}n-Zygmund operator, commutator, space $\mathrm{BMO}$, Lipschitz space, generalized variable exponent Morrey space

MSC numbers: Primary 42B20, 42B25, 42B35

Supported by: This work was financially supported by Young Teachers' Scientific Research Ability Promotion Project of Northwest Normal University (NWNU-LKQN2020-07), Innovation Fund Project for Higher Education of Gansu Province (2020A-010) and NNSF(11561062).