BOUNDEDNESS OF BILINEAR INTEGRAL OPERATORS RELATED TO GENERALIZED KERNELS AND THEIR COMMUTATORS ON PRODUCT OF GENERALIZED VARIABLE EXPONENTS MORREY SPACES

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.