Endpoint bounds for multilinear fractional integrals

We prove that the multilinear fractional integral operator$I _{\alpha} (f_1,\ldots, f_k )(x) = \int_{\mathbb{R}^n } f_1 ( x - \theta _1 y ) \ldots f_k ( x -\theta _k y ) | y |^{\alpha - n} dy$, where $\theta _j, \ j=1, \ldots , k$ are distinct and nonzero, (due toGrafakos) has the endpoint weak-type boundednessinto $ L^{r ,\infty}$ when $r = \frac{n}{2n -\alpha}$. Hence, we obtain by the multilinearinterpolation theorem that $ I _{\alpha}$ is boundedinto $ L^r$ for all $ r > \frac{n}{2n - \alpha}$.Moreover, We also prove that $ I_{\alpha}$ is notbounded into $ L^r $ for any $r < \frac{n}{2n - \alpha}$ under some conditions on $ \theta _j $'s. Similarly, weshow that the multilinear Hilbert transform $ H (f, g, h_1, \ldots , h_k) (x) = \textrm{ p.v.} \int f (x+t) g (x-t)\prod_{ j=1} ^k h_j (x - \theta _j t) \frac{dt}{t}$, where $ \theta _j \neq \pm 1$ are distinct and nonzero,is not bounded into $L^r$ for any $r<\frac{1}{2}$under some conditions on $\theta_j$'s.

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Published 15 March 2013