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Multilinear estimates and fractional integration. (English) Zbl 0952.42005
The multilinear operator on $L^{p_1}\times\ldots\times L^{p_{k+1}}$ is defined by a kernel $K$ as $$T(f_{p_1},\ldots,f_{p_{k+1}})= \int f_1(l_1)\ldots f_{k+1}(l_{k+1}) K(x_1,\ldots,x_k) dx_1\ldots dx_k,$$ where $l_j$ are linear mappings. The authors consider the case of the fractional integral with the kernel $K(x)=\frac{1}{|x|^{-nk+\alpha}}$ and prove that this operator is bounded from $L^{p_1}\times\ldots\times L^{p_{k+1}}$ to $L^q$, where $\frac{1}{q}=\frac{1}{p_1}+\ldots +\frac{1}{p_{k+1}}- \frac{\alpha}{n}$, when all $p_j>1$ and weakly bounded for some $p_j$ equal 1. The analogous result is presented for Calderón-Zygmund kernels, which extends the result of {\it R. R. Coifman} and {\it Y. Meyer} [Trans. Am. Math. Soc. 212, 315-331 (1975; Zbl 0324.44005)] to the case when the exponent $q$ is less then $1$.

42B20Singular and oscillatory integrals, several variables
47G10Integral operators
26A33Fractional derivatives and integrals (real functions)
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