Yuan Zhao, Haibo Lin, Yan Meng. Source: Annals of Functional Analysis, Volume 10, Number 3, 337--349.Abstract:
Let $({\mathcal{X}},d,\mu )$ be a metric measure space such that, for any fixed $x\in {\mathcal{X}}$ , $\mu (B(x,r))$ 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 $L^{1}(\mu )\times \cdots \times L^{1}(\mu )\times L^{p_{k+1}}(\mu )\times\cdots \times L^{p_{m}}(\mu )$ into the Lebesgue space $L^{q}(\mu )$ , where $k\in [1,m)\cap {\mathbb{N}}$ , $\alpha \in [k,m)$ , $p_{i}\in (1,\infty )$ for $i\in \{k+1,\ldots ,m\}$ and $1/q=k+\sum _{i=k+1}^{m}1/{p_{i}}-\alpha $ . We furthermore prove that $I_{m,\alpha }$ is bounded from $L^{p_{1}}(\mu )\times \cdots \times L^{p_{m}}(\mu )$ into $L^{\infty }(\mu )$ , where $p_{i}\in (1,\infty )$ for $i\in \{1,\ldots ,m\}$ and $\sum _{i=1}^{m}1/{p_{i}}=\alpha \in [1,m)$ .