Some Estimates for Multilinear Calder\'{o}n-Zygmund Operators with Fractional Kernels of Dini's Type and their Commutators on Generalized Fractional Mixed Morrey Spaces

Abstract

In this paper, we mainly study the boundedness of multilinear Calder\'{o}n-Zygmund operator with fractional kernel of type $h(t)$ and its commutators on generalized fractional mixed Morrey space $L^{\vec{q},\eta,\psi}(\mathbb{R}^{n})$. Firstly, with the help of the extrapolation theorem, the monotone convergence theorem and the boundedness of fractional integral operator $I_{\beta}$ on mixed Lebesgue space $L^{\vec{q}}(\mathbb{R}^{n})$, we obtain the boundedness of $T$ on space $L^{\vec{q}}(\mathbb{R}^{n})$. Secondly, the boundedness of $T$ on space $L^{\vec{q},\eta,\psi}(\mathbb{R}^{n})$ is derived by applying the boundedness of $T$ on space $L^{\vec{q}}(\mathbb{R}^{n})$. Thirdly, we prove that the commutators $T_{\prod{\vec{b}}}$ and $T_{\sum{\vec{b}}}$ are bounded from $L^{\vec{p_{1}},\eta_{1},\psi}(\mathbb{R}^{n})\times\cdots\times{L^{\vec{p_{m}},\eta_{m},\psi}(\mathbb{R}^{n})}$ to $L^{\vec{q},\eta,\psi}(\mathbb{R}^{n})$. Finally, as applications, the boundedness for the multilinear fractional new maximal operator $\mathcal{M}_{\varphi,\beta}$ and its commutators $\mathcal{M}_{\vec{b},\varphi,\beta}$ and $[\vec{b},\mathcal{M}_{\varphi,\beta}]$ on space $L^{\vec{q},\eta,\psi}(\mathbb{R}^{n})$ is presented.