We prove $L^p$ bounds for the extensions of standard multilinear Calderon-Zygmund operators to tuples of UMD spaces tied by a natural product structure. This can, for instance, mean the pointwise product in UMD function lattices, or the composition of operators in the Schatten-von Neumann subclass of the algebra of bounded operators on a Hilbert space. We do not require additional assumptions beyond UMD on each space - in contrast to previous results, we e.g. show that the Rademacher maximal function property is not necessary. The obtained generality allows for novel applications. For instance, we prove new versions of fractional Leibniz rules via our results concerning the boundedness of multilinear singular integrals in non-commutative $L^p$ spaces. Our proof techniques combine a novel scheme of induction on the multilinearity index with dyadic-probabilistic techniques in the UMD space setting.

中文翻译：

---

非交换 $$L^p$$ 空间上的多线性奇异积分

我们证明了标准多线性 Calderon-Zygmund 算子扩展到由自然乘积结构绑定的 UMD 空间元组的 $L^p$ 边界。例如，这可以表示 UMD 函数格中的逐点积，或者 Hilbert 空间上有界算子代数的 Schatten-von Neumann 子类中算子的组合。我们不需要对每个空间的 UMD 之外的额外假设 - 与之前的结果相反，我们例如表明 Rademacher 极大函数属性不是必需的。获得的普遍性允许新的应用。例如，我们通过关于非交换 $L^p$ 空间中多重线性奇异积分的有界性的结果证明了分数莱布尼茨规则的新版本。