We prove that the class of trilinear multiplier forms with singularity over a one dimensional subspace, including the bilinear Hilbert transform, admit bounded $L^p$-extension to triples of intermediate $\mathrm{UMD}$ spaces. No other assumption, for instance of Rademacher maximal function type, is made on the triple of $\mathrm{UMD}$ spaces. Among the novelties in our analysis is an extension of the phase-space projection technique to the $\mathrm{UMD}$-valued setting. This is then employed to obtain appropriate single tree estimates by appealing to the $\mathrm{UMD}$-valued bound for bilinear Calderon-Zygmund operators recently obtained by the same authors.

中文翻译：

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具有调制不变性的 Banach 值多重线性奇异积分

我们证明了一类在一维子空间上具有奇点的三线性乘子形式，包括双线性希尔伯特变换，承认有界 $L^p$-扩展到中间 $\mathrm{UMD}$ 空间的三元组。在 $\mathrm{UMD}$ 空间的三元组上没有其他假设，例如 Rademacher 极大函数类型。我们分析中的新颖之处在于将相空间投影技术扩展到 $\mathrm{UMD}$ 值设置。然后，通过利用同一作者最近获得的双线性 Calderon-Zygmund 算子的 $\mathrm{UMD}$ 值界限，将其用于获得适当的单树估计。