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arXiv 1902.04527math.CA
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多线性分数次积分算子在具有混合范数的Lebesgue空间上到线性算子的推广

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

Ting Chen, Wenchang Sun

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74 pages
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摘要：

在[C. e. Kenig和E. M. Stein，多线性估计和分数阶积分，Math.Res.Lett.，6（1）：1 - 15，1999]，研究了以下类型的多线性分数次积分\[ \int_{\mathbb{R}^{mn}} \frac{f_1(l_1(x_1,\ldots,x_m,x))\cdots f_{m+1}(l_{m+1}(x_1,\ldots,x_m,x))}{(|x_1|+\ldots+|x_m|)^λ} dx_1\ldots dx_m \]，其中$l_i$是满足一定条件的从$\mathbb{R}^{(m+1)n}$到$\mathbb{R}^n$的线性映射.证明了当指标满足齐性条件时，从$L^{p_1}\times \ldots \times L^{p_{m+1}}$到$L^q$的多线性分数次积分的有界性。在本文中，我们证明了上述多线性分数次积分扩展到一个线性算子的混合范数Lebesgue空间$L^{\vec p}$，其中包含$L^{p_1}\times \ldots \times L^{p_{m+1}}$作为一个子集的功能。在对线性映射$l_i$的限制较少的情况下，给出了算子从$L^{\vec p}$到$L^q$有界的指标$\vec p$，$q$和$λ$的完全刻画.对于$m=1$或$n=1$，给出了$(l_1, \ldots, l_{m+1})$，$\vec p=(p_1,\ldots, p_{m+1})$，$q$和$λ$有界的充要条件.

英文摘要：

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 \[ \int_{\mathbb{R}^{mn}} \frac{f_1(l_1(x_1,\ldots,x_m,x))\cdots f_{m+1}(l_{m+1}(x_1,\ldots,x_m,x))}{(|x_1|+\ldots+|x_m|)^λ} dx_1\ldots dx_m \] 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 $λ$ 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 $λ$ such that the operator is bounded.

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