Boundedness of Bilinear Integral Operators Related to Generalized Kernels and Their Commutators on Product of Generalized Variable Exponents Morrey Spaces
Year: 2025
Type: article
Abstract: The purpose of this paper is to investigate the boundedness of a bilinear integral operator $ T $ and its commutator $ T_{b_{1},b_{2}} $ formed by $ b_{1}, b_{2}\in\mathrm{BMO}(\mathbb{R}^{n}) $ and the $ T $ on product of generalized variable Morrey spaces $ \mathcal{M}^{p_{1}(\cdot),\varphi_{1}}(\mathbb{R}^{n})\times\mathcal{M}^{p_{2}(\cdot),\varphi_{2}}(\mathbb{R}^{n}) $. Under assumption that Lebesgue measurable functions $ \varphi_{i} $ belong to the class $ \mathbb{W}_{p_{i}(\cdot)} $, $ i=1,2 $, the authors prove that the $ T $ is bounded from product of spaces $ \mathcal{M}^{p_{1}(\cdot),\varphi_{1}}(\mathbb{R}^{n})\times\mathcal{M}^{p_{2}(\cdot),\varphi_{2}}(\mathbb{R}^{n}) $ into spaces $ \mathcal{M}^{p(\cdot),\varphi}(\mathbb{R}^{n}) $; furthermore, the authors show that the $ T_{b_{1},b_{2}} $ is bounded from product of spaces $ \mathcal{M}^{p_{1}(\cdot),\varphi_{1}}(\mathbb{R}^{n}) \times\mathcal{M}^{p_{2}(\cdot),\varphi_{2}}(\mathbb{R}^{n}) $ into spaces $ \mathcal{M}^{p(\cdot),\varphi}(\mathbb{R}^{n}) $, where $ \varphi_{1}\varphi_{2}=\varphi $, and $ \frac{1}{p(\cdot)}=\frac{1}{p_{1}(\cdot)}+\frac{1}{p_{2}(\cdot)} $ with $ p_{1}(\cdot), p_{2}(\cdot)\in\mathcal{P}(\mathbb{R}^{n}) $. As corollaries, the boundedness of the $ T $ and $ T_{b_{1},b_{2}} $ on product of variable exponent Morrey spaces $ L^{p_{1}(\cdot),\kappa}(\mathbb{R}^{n})\times L^{p_{2}(\cdot),\kappa}(\mathbb{R}^{n}) $ is established, respectively.
DOI: 10.11948/20250045
Authors: Miaomiao Wang, Guanghui Lu
Institution: Northwest Normal University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
Award: G6353860794
OpenAlex ID: W7107947804