We develop a general theory of multilinear singular integrals with operator-valued kernels, acting on tuples of UMD Banach spaces. This, in particular, involves investigating multilinear variants of the $\mathcal R$-boundedness condition naturally arising in operator-valued theory. We proceed by establishing a suitable representation of multilinear, operator-valued singular integrals in terms of operator-valued dyadic shifts and paraproducts, and studying the boundedness of these model operators via dyadic-probabilistic Banach space-valued analysis. In the bilinear case, we obtain a $T(1)$-type theorem without any additional assumptions on the Banach spaces other than the necessary UMD. Higher degrees of multilinearity are tackled via a new formulation of the Rademacher maximal function (RMF) condition. In addition to the natural UMD lattice cases, our RMF condition covers suitable tuples of non-commutative $L^p$-spaces. We employ our operator-valued theory to obtain new multilinear, multi-parameter, operator-valued theorems in the natural setting of UMD spaces with property $\alpha$.

中文翻译：

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多线性算子值 Calderón-Zygmund 理论

我们开发了具有算子值内核的多重线性奇异积分的一般理论，作用于 UMD Banach 空间的元组。这尤其涉及研究在算子值理论中自然出现的 $\mathcal R$ 有界条件的多线性变体。我们继续根据算子值二进位移和副积建立多线性、算子值奇异积分的合适表示，并通过二进概率 Banach 空间值分析研究这些模型算子的有界性。在双线性情况下，我们获得了一个 $T(1)$ 类型的定理，除了必要的 UMD 之外，对 Banach 空间没有任何额外的假设。通过 Rademacher 极大函数 (RMF) 条件的新公式来解决更高程度的多重线性问题。除了自然的 UMD 格子情况外，我们的 RMF 条件涵盖了非交换 $L^p$-spaces 的合适元组。我们使用我们的算子值理论在具有属性 $\alpha$ 的 UMD 空间的自然设置中获得新的多线性、多参数、算子值定理。