一类半单李群上径向$L^p$乘子的Sobolev估计
A Sobolev estimate for radial $L^p$-multipliers on a class of semi-simple Lie groups
AI总结：
本文针对Harish-Chandra类半单李群，建立了径向$L^p$乘子的Sobolev估计，给出了乘子有界性的条件，并得到衰减率随$p$趋近于2而变慢的新$L^p$-Fourier乘子例子。
摘要：
设$G$为Harish-Chandra类中的半单李群，其极大紧子群为$K$。设$Ω_K$为径向Casimir算子的负算子。令$\frac{1}{4} \dim(G/K) < S_G < \frac{1}{2} \dim(G/K)$，$s \in (0, S_G]$且$p \in (1,\infty)$满足$\left| \frac{1}{p} - \frac{1}{2} \right| < \frac{s}{2 S_G}$。则存在常数$C_{G,s,p} >0$，使得对每个双$K$-不变的$m \in L^\infty(G) \cap L^2(G)$，若$m \in {\rm Dom}(Ω_K^s)$且$Ω_K^s(m) \in L^{2S_G/s}(G)$，则有$\Vert T_m: L^p(\widehat{G}) \rightarrow L^p(\widehat{G}) \Vert \leq C_{G, s,p} \Vert Ω_K^s(m) \Vert_{L^{2S_G/s}(G)}$，其中$T_m$是符号为$m$的Fourier乘子，作用于$G$的群von Neumann代数的非交换$L^p$空间。这给出了新的$L^p$-Fourier乘子例子，其衰减率随$p$趋近于2而变慢。
英文摘要：
Let $G$ be a semi-simple Lie group in the Harish-Chandra class with maximal compact subgroup $K$. Let $Ω_K$ be minus the radial Casimir operator. Let $\frac{1}{4} \dim(G/K) < S_G < \frac{1}{2} \dim(G/K) , s \in (0, S_G]$ and $p \in (1,\infty)$ be such that \[ \left| \frac{1}{p} - \frac{1}{2} \right| < \frac{s}{2 S_G}. \] Then, there exists a constant $C_{G,s,p} >0$ such that for every $m \in L^\infty(G) \cap L^2(G)$ bi-$K$-invariant with $m \in {\rm Dom}(Ω_K^s)$ and $Ω_K^s(m) \in L^{2S_G/s}(G)$ we have, \[ \Vert T_m: L^p(\widehat{G}) \rightarrow L^p(\widehat{G}) \Vert \leq C_{G, s,p} \Vert Ω_K^s(m) \Vert_{L^{2S_G/s}(G)}, \] where $T_m$ is the Fourier multiplier with symbol $m$ acting on the non-commutative $L^p$-space of the group von Neumann algebra of $G$. This gives new examples of $L^p$-Fourier multipliers with decay rates becoming slower when $p$ approximates $2$.
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