With the invitation of Associate Professor Sibei Yang of the school of mathematics and statistics of Lanzhou University, Professor Wenchang Sun of Department of Mathematics, NankaiUniversity, will visit our school from December 7, 2019 to December 9, 2019 to conduct academic exchanges and make an academic report.

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

Time：10:00 a.m.，December 9, 2019

Location：Qiyun building, Room 911

Abstract: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

$$\int_{\mathbbR^{mn}}\frac{f_1(l_1(x_1,\ldots,x_m,x))\cdotsf_{m+1}(l_{m+1}(x_1,\ldots,x_m,x))}{(|x_1|+\ldots+|x_m|)^{\lambda}} dx_1\ldots dx_m$$

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 talk, 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.