Multilinear Riesz potential operators on Herz-type spaces and generalized Morrey spaces

Abstract

Let $m, n$ be integers with $n\geq 2$, $m\geq 1$, the multilinear Riesz potential operators be defined by $$ I_{\alpha}^{(m)}({\bf{f}})(x) = \int_{(\mathbb{R}^{n})^{m}} \frac{f_1(y_1) \dots f_m(y_m)}{| (x-y_1, \dots, x-y_m) |^{mn-\alpha}}d{\bf{y}}, $$ where ${\bf{y}}=(y_1, \dots, y_m)$ and ${\bf{f}}=(f_{1}, \dots, f_{m})$. In the first part of this paper, the boundedness for the operator $I_{\alpha}^{(m)}$ on the homogeneous Herz-Morrey product spaces, $M\dot{K}_{p_1,q_1}^{n(1-1/q_1),\lambda_1}(\mathbb{R}^n) \times\dots\times M\dot{K}_{p_m,q_m}^{n(1-1/q_m),\lambda_m}(\mathbb{R}^n)$, and on the Herz-type Hardy product spaces, $H\dot{K}_{q_1}^{\sigma_1,p_1}(\mathbb{R}^n) \times\dots\times H\dot{K}_{q_m}^{\sigma_m,p_m}(\mathbb{R}^n)$ for $\sigma_i>n(1-1/q_i)$, are established respectively. The second goal of the paper is to extend the known $L^p$-bounded\-ness of $I_\alpha^{(m)}$ to generalized Morrey spaces, $L^{p,\phi}(\mathbb{R}^n)$, where $p\in[1,+\infty)$ and $\phi$ is the suitable doubling and integral functions.