×

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

Citations:

Zbl 0324.44005
PDF BibTeX XML Cite
Full Text: DOI