论文标题
奇异空间中的不确定性估计和节点集的大小
Indeterminacy estimates and the size of nodal sets in singular spaces
论文作者
论文摘要
我们获得了最近在[47]中引入的不确定性原理的尖锐版本，并通过[13]进行了改进，将连续函数的零集的尺寸与零均值的零集和正质量质量之间的最佳运输成本相关联。结果实际上对于验证RICCI曲率上的合成下限的广泛的度量度量空间，即MCP（K，N）或CD（K，N）条件，因此也将范围扩展到Riemannian歧管平滑环境之外。将不确定性原理应用于可能是非平滑空间的Laplacian的本征函数，我们根据特征值获得了其淋巴结集的大小的新界限。那些可能是非线性的那些情况，也涵盖了laplacian本征征的线性组合的应用。据我们所知，以前没有以非平滑空间为名的结果。
We obtain the sharp version of the uncertainty principle recently introduced in [47], and improved by [13], relating the size of the zero set of a continuous function having zero mean and the optimal transport cost between the mass of the positive part and the negative one. The result is actually valid for the wide family of metric measure spaces verifying a synthetic lower bound on the Ricci curvature, namely the MCP(K,N) or CD(K,N) condition, thus also extending the scope beyond the smooth setting of Riemannian manifolds. Applying the uncertainty principle to eigenfunctions of the Laplacian in possibly non-smooth spaces, we obtain new lower bounds on the size of their nodal sets in terms of the eigenvalues. Those cases where the Laplacian is possibly non-linear are also covered and applications to linear combinations of eigenfunctions of the Laplacian are derived. To the best of our knowledge, no previous results were known for non-smooth spaces.
