Heisenberg–Pauli–Weyl uncertainty inequalities and polynomial volume growth
Heisenberg–Pauli–Weyl uncertainty inequalities and polynomial volume growth
复制标题
DOI：
10.1016/j.aim.2007.03.014
复制
发表时间：
2007-11
影响因子：
1.7
通讯作者：
P. Ciatti;F. Ricci;M. Sundari
中科院分区：
文献类型：
--
作者：
P. Ciatti;F. Ricci;M. Sundari
在其更简单的形式中，Heisenberg-Pauli-Weyl不等式说，在本文中，我们将这个不等式推广到具有“规范函数”的度量空间上的正自伴算子L，使得(A)球的度量由半径的幂控制(对于大小球可能是不同的幂)；(B)L生成的半群满足具有相同类型多项式界的超紧估计。我们给出了这一结果在多项式体积增长群上的次拉普拉斯算子和幂零群上的某些高阶左不变亚椭圆算子上的应用的例子。最后，我们证明了这些估计还隐含了局部不确定不等式的推广形式。
In its simpler form, the Heisenberg–Pauli–Weyl inequality says that In this paper, we extend this inequality to positive self-adjoint operators L on measure spaces with a “gauge function” such that (a) measures of balls are controlled by powers of the radius (possibly different powers for large and small balls); (b) the semigroup generated by L satisfies ultracontractive estimates with polynomial bounds of the same type. We give examples of applications of this result to sub-Laplacians on groups of polynomial volume growth and to certain higher-order left-invariant hypoelliptic operators on nilpotent groups. We finally show that these estimates also imply generalized forms of local uncertainty inequalities.