Mathematical Research Letters

Volume 19 (2012)

Endpoint bounds for multilinear fractional integrals

Pages: 1145 – 1154

DOI: http://dx.doi.org/10.4310/MRL.2012.v19.n5.a15

Authors

Seungwoo Kuk (Department of Mathematical Sciences, KAIST, Daejeon, South Korea)

Sungyun Lee (Department of Mathematical Sciences, KAIST, Daejeon, South Korea)

Abstract

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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