## Multilinear estimates and fractional integration.(English)Zbl 0952.42005

Math. Res. Lett. 6, No. 1, 1-15 (1999); erratum ibid. 6, No. 3-4, 467 (1999).
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 R. R. Coifman and Y. Meyer [Trans. Am. Math. Soc. 212, 315-331 (1975; Zbl 0324.44005)] to the case when the exponent $$q$$ is less then $$1$$.

### MSC:

 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 47G10 Integral operators 26A33 Fractional derivatives and integrals

Zbl 0324.44005
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