局部紧群上的$L^p$ - $L^q$乘子
$L^p$-$L^q$ multipliers on locally compact groups
摘要：
本文讨论了一般局部紧可分么模群$G$上的谱乘子和Fourier乘子在值域$1<p\leq q<\infty$上的$L^p$-$L^q$有界性.证明了$1<p\leq q<\infty$的Lizorkin型乘子定理，并将其改进为$1<p\leq 2\leq q<\infty$的Hörmander型乘子定理.在此过程中，我们建立了一般局部紧可分幺模群上的Paley不等式和Hausdorff-Young-Paley不等式。作为Hörmander型乘子定理的结果，我们得到一般局部紧可分幺模群上的谱乘子定理。然后我们应用它得到嵌入定理以及$G$上一般正无界不变算子的热核的$L^p$ - $L^q$范数的时间渐近性。我们说明了所得到的结果，紧李群和海森堡群上的次拉普拉斯算子。我们发现，我们的结果意味着已知的结果$L^p$ - $L^q$乘子，如Hörmander的傅立叶乘子定理$\mathbb{R}^{n}$或已知的结果紧李群上的傅立叶乘子。在本文中开发的新方法依赖于分析组冯诺依曼代数所需的乘数定理的推导。
英文摘要：
In this paper we discuss the $L^p$-$L^q$ boundedness of both spectral and Fourier multipliers on general locally compact separable unimodular groups $G$ for the range $1<p\leq q<\infty$. We prove a Lizorkin type multiplier theorem for $1<p\leq q<\infty$, and then refine it as a Hörmander type multiplier theorem for $1<p\leq 2\leq q<\infty$. In the process, we establish versions of Paley and Hausdorff-Young-Paley inequalities on general locally compact separable unimodular groups. As a consequence of the Hörmander type multiplier theorem we derive a spectral multiplier theorem on general locally compact separable unimodular groups. We then apply it to obtain embedding theorems as well as time-asymptotics for the $L^p$-$L^q$ norms of the heat kernels for general positive unbounded invariant operators on $G$. We illustrate the obtained results for sub-Laplacians on compact Lie groups and on the Heisenberg group. We show that our results imply the known results for $L^p$-$L^q$ multipliers such as Hörmander's Fourier multiplier theorem on $\mathbb{R}^{n}$ or known results for Fourier multipliers on compact Lie groups. The new approach developed in this paper relies on the analysis in the group von Neumann algebra for the derivation of the desired multiplier theorems.
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