# "九章讲坛"第193讲 — 孙文昌 教授

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 satisfythe homogeneity condition.

In this talk, we show that the above multilinear fractional integral extends to a linear operatorfor 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 operatoris bounded.

2019年12月6日