Continuities of bilinear singular integral operators associated with generalized kernels on product mixed generalized Morrey spaces

다국어 초록 (Multilingual Abstract)

The aim of this paper is to establish the boundedness of a bilinear singular integral operator $\widetilde{T}$ associated with generalized kernels and its corresponding commutator $\widetilde{T}_{b_{1},b_{2}}$ formed by $b_{1}, b_{2}\in\mathrm{BMO}(\mathbb{R}^{n})$ and $\widetilde{T}$ on product mixed Lebesgue spaces $L^{\vec{p}}(\mathbb{R}^{n})$ and product mixed generalized Morrey spaces $\mathcal{L}^{u}_{\vec{p}}(\mathbb{R}^{n})$. Via some known results for the operators $\widetilde{T}$ and $\widetilde{T}_{b_{1},b_{2}}$, the authors prove that $\widetilde{T}$ and $\widetilde{T}_{b_{1},b_{2}}$ are bounded from product spaces $L^{\vec{p}_{1}}(\mathbb{R}^{n})\times L^{\vec{p}_{2}}(\mathbb{R}^{n})$ into spaces $L^{\vec{p}}(\mathbb{R}^{n})$. Furthermore, under assumption that Lebesgue measurable functions $u_{1}, u_{2}$ and $u$ belong to the class $\mathbb{W}_{\vec{p}}$ and meet $u_{1}u_{2}=u$, the authors show that $\widetilde{T}$ and $\widetilde{T}_{b_{1},b_{2}}$ are bounded from product spaces $\mathcal{L}^{u_{1}}_{\vec{p}_{1}}(\mathbb{R}^{n})\times \mathcal{L}^{u_{2}}_{\vec{p}_{2}}(\mathbb{R}^{n})$ into spaces $\mathcal{L}^{u}_{\vec{p}}(\mathbb{R}^{n})$, respectively.