Bilinear Strongly Singular Calder\'{o}n-Zygmund Operators and Their Commutators on Non-Homogeneous Generalized Morrey Spaces

Abstract:

      The main goal of this paper is to establish the boundedness of bilinear strongly singular operator $\widetilde{T}$ and its commutator $\widetilde{T}_{b_{1},b_{2}}$ on generalized Morrey spaces $M^{u}_{p}(\mu)$ over non-homogeneous metric measure spaces. Under assumption that the Lebesgue measurable functions $u, u_{1}$ and $u_{2}$ belong to $\mathbb{W}_{\tau}$ for $\tau\in(0,2)$, and $u_{1}u_{2}=u$. The authors prove that $\widetilde{T}$ is bounded from product spaces $M^{u_{1}}_{p_{1}}(\mu)\times M^{u_{2}}_{p_{2}}(\mu)$ into spaces $M^{u}_{p}(\mu)$, where $\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$ with $1