We consider a multilinear kernel operator between Banach function spaces over finite measures and suitable order continuity properties, namely $T:X_{1}(\,\unicode[STIX]{x1D707}_{1})\times \cdots \times X_{n}(\,\unicode[STIX]{x1D707}_{n})\rightarrow Y(\,\unicode[STIX]{x1D707}_{0})$. Then we define, via duality, a class of linear operators associated to the $j$-transpose operators. We show that, under certain conditions of $p$th power factorability of such operators, there exist vector measures $m_{j}$ for $j=0,1,\ldots ,n$ so that $T$ factors through a multilinear operator $\widetilde{T}:L^{p_{1}}(m_{1})\times \cdots \times L^{p_{n}}(m_{n})\rightarrow L^{p_{0}^{\prime }}(m_{0})^{\ast }$, provided that $1/p_{0}=1/p_{1}+\cdots +1/p_{n}$. We apply this scheme to the study of the class of multilinear Calderón–Zygmund operators and provide some concrete examples for the homogeneous polynomial and multilinear Volterra and Laplace operators.

中文翻译：

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通过向量度量空间分解多线性内核运算符

我们考虑了有限度量上的 Banach 函数空间和合适的阶连续性属性之间的多线性核算子，即$T:X_{1}(\,\unicode[STIX]{x1D707}_{1})\times \cdots \times X_{n}(\,\unicode[STIX]{x1D707}_{n})\右箭头 Y(\,\unicode[STIX]{x1D707}_{0})$. 然后我们通过对偶定义一类与$j$-转置运算符。我们证明，在特定条件下$p$此类算子的功率因数，存在矢量度量$m_{j}$为了$j=0,1,\ldots ,n$以便$T$通过多线性算子的因子$\widetilde{T}:L^{p_{1}}(m_{1})\times \cdots \times L^{p_{n}}(m_{n})\rightarrow L^{p_{0} ^{\prime }}(m_{0})^{\ast }$, 前提是$1/p_{0}=1/p_{1}+\cdots +1/p_{n}$. 我们将该方案应用于多线性 Calderón–Zygmund 算子类的研究，并为齐次多项式和多线性 Volterra 和 Laplace 算子提供了一些具体示例。