Extension of Multilinear Fractional Integral Operators to Linear Operators on Lebesgue Spaces with Mixed Norms

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 $${\int }_{{\mathbb{R}}^{mn}}\frac{f_1(l_1(x_1,\dots ,x_m,x))\cdots f_{m+1}(l_{m+1}(x_1,\dots ,x_m,x))}{(\mathrm{\mid }{x}_1\mathrm{\mid }+\dots +\mathrm{\mid }{x}_m\mathrm{\mid }{)}^{\lambda }}d{x}_1\dots d{x}_m$$ 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 $L^{p_1}\times \dots \times L^{p_{m+1}}$ 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 $(l_1,\dots ,l_{m+1})$, $\vec p=(p_1,\ldots, p_{m+1})$,$q$and$\lambda$ such that the operator is bounded.