In [C. E. Kenig and E. M. Stein, Multilinear estimates and fractional integration, Math. Res. Lett., 6(1):1-15, 1999], the following type of multilinear fractional integral \[ \int_{ \mathbb{ R}^{ mn}} \frac{ f_1(l_1(x_1,\ldots,x_m,x))\cdots f_{ m+1}(l_{ m+1}(x_1,\ldots,x_m,x))}{ (|x_1|+\ldots+|x_m|)^{ \lambda}} dx_1\ldots dx_m \] was studied, where $l_i$ are linear maps from $\mathbb{ R}^{ (m+1)n}$ to $\mathbb{ R}^n$ satisfying certain conditions. They proved the boundedness of such multilinear fractional integral from $L^{ p_1}\times \ldots \times L^{ p_{ m+1}}$ to $L^q$ when the indices satisfy the homogeneity condition. In this paper, we show that the above multilinear fractional integral extends to a linear operator for functions in the mixed-norm Lebesgue space $L^{ \vec p}$ which contains $L^{ p_1}\times \ldots \times L^{ p_{ m+1}}$ as a subset. Under less restrictions on the linear maps $l_i$, we give a complete characterization of the indices $\vec p$, $q$ and $\lambda$ for which such an operator is bounded from $L^{ \vec p}$ to $L^q$. And for $m=1$ or $n=1$, we give necessary and sufficient conditions on $(l_1, \ldots, l_{ m+1})$, $\vec p=(p_1,\ldots, p_{ m+1})$, $q$ and $\lambda$ such that the operator is bounded.
Categories: math.CA