We introduce the class of Riesz-representable multilinear mappings on products of $L_1\left(\mu \right)$ spaces for finite measures μ and give their main properties. We prove Grothendieck type theorems for composition with the natural operators $L_{\mathrm{\infty }}\to L_1$ and $C\left(\mathrm{\Omega }\right)\to L_1$. Unlike the linear case, the composition is not nuclear but is characterized by what we call the left integral ${\ell }_1$-factorization property.

中文翻译：

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Riesz 可表示的多线性映射

我们在乘积上引入 Riesz 可表示的多线性映射类${\mathrm{大号}}_1\left(\mu \right)$有限量度μ的空间并给出它们的主要性质。我们用自然算子证明了格罗腾迪克类型定理${\mathrm{大号}}_{\mathrm{\infty }}\to {\mathrm{大号}}_1$和$C\left(\mathrm{\Omega }\right)\to {\mathrm{大号}}_1$. 与线性情况不同，合成不是核的，而是由我们所说的左积分来表征${\ell }_1$-因式分解性质。