论文标题
轨道对称空间上轨道测量的平稳性
The smoothness of orbital measures on noncompact symmetric spaces
论文作者
论文摘要
令$ g/k $是一个不可约的对称空间，其中$ g $是一个非紧缩的，连接的谎言组，$ k $是一个紧凑的连接子组。我们使用球形函数的衰减属性表明，任何$ r = r（g/k）$连续轨道测量的卷积产物具有其密度函数的$％l^{2}（g）$，因此对于HAAR度量是绝对连续的措施。数字$ r $大约是$ g/k $的排名。对于轨道测量的特殊情况，$ν_{a_ {i}} $，在双cosets $ ka_ {i} k $上支持，其中$ a_ {i {i} $属于常规元素的密集集，我们证明了$ν_{a _ _ _ {1}}}} \ ast ast a al ast cart l^{2}，$除了需要三个轨道测量的卷积时，cartan type $ ai $的对称空间（即使$ν_{a_ {1}} \ astastν_{a_ {a_ {2}} $绝对连续）。
Let $G/K$ be an irreducible symmetric space where $G$ is a non-compact, connected Lie group and $K$ is a compact, connected subgroup. We use decay properties of the spherical functions to show that the convolution product of any $r=r(G/K)$ continuous orbital measures has its density function in $% L^{2}(G)$ and hence is an absolutely continuous measure with respect to Haar measure. The number $r$ is approximately the rank of $G/K$. For the special case of the orbital measures, $ν_{a_{i}}$, supported on the double cosets $Ka_{i}K$ where $a_{i}$ belongs to the dense set of regular elements, we prove the sharp result that $ν_{a_{1}}\ast ν_{a_{2}}\in L^{2},$ except for the symmetric space of Cartan type $AI$ when the convolution of three orbital measures is needed (even though $ν_{a_{1}}\ast ν_{a_{2}}$ is absolutely continuous).
