The multilinear operator on is defined by a kernel as
where are linear mappings.
The authors consider the case of the fractional integral with the kernel and prove that this operator is bounded from to , where , when all and weakly bounded for some 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 is less then 1.