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FACTORIZING MULTILINEAR KERNEL OPERATORS THROUGH SPACES OF VECTOR MEASURES
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 112 / Issue 1 / February 2022
- Published online by Cambridge University Press:
- 13 January 2020, pp. 30-51
- Print publication:
- February 2022
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- Article
- Export citation
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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.
