Endpoint estimates for multilinear fractional integral operators on metric measure spaces

Abstract

Let $(\mathcal{X},d,\mu )$ be a metric measure space such that, for any fixed $x\in \mathcal{X}$, $\mu \left(B\right(x,r\left)\right)$ is a continuous function with respect to $r\in (0,\infty )$. In this paper, we prove endpoint estimates for the multilinear fractional integral operators $I_{m,\alpha }$ from the product of Lebesgue spaces $L^1\left(\mu \right)\times \cdots \times L^1\left(\mu \right)\times L^{p_{k+1}}\left(\mu \right)\times \cdots \times L^{p_m}\left(\mu \right)$ into the Lebesgue space $L^q\left(\mu \right)$, where $k\in [1,m)\cap \mathbb{N}$, $\alpha \in [k,m)$, $p_i\in (1,\infty )$ for $i\in \{k+1,\dots ,m\}$ and $1/q=k+{\sum }_{i=k+1}^{m}1/p_i-\alpha$. We furthermore prove that $I_{m,\alpha }$ is bounded from $L^{p_1}\left(\mu \right)\times \cdots \times L^{p_m}\left(\mu \right)$ into $L^{\infty }\left(\mu \right)$, where $p_i\in (1,\infty )$ for $i\in \{1,\dots ,m\}$ and ${\sum }_{i=1}^{m}1/p_i=\alpha \in [1,m)$.