各向同性Grassmann和Heisenberg群的投影相交和切片定理

IF 0.9 3区数学 Q2 MATHEMATICS

Analysis and Geometry in Metric Spaces Pub Date : 2019-07-16 DOI:10.1515/agms-2020-0002

Fernando Roman-Garcia

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引用次数: 0

摘要

摘要本文研究了的子集的各向同性投影相交的Hausdorff维数ℝ2n以及集合与各向同性平面的交点的维数。证明了如果A和B是ℝ2n，则对于各向同性m平面的正测度集，在正交投影到这些平面上的a和B的图像的交集具有正Hausdorff m测度。此外，如果A是Hausdorff维数大于m的可测量集合，则存在集合B⊂ℝ2n与dim B⩽m使得对于所有x∈ℝ2n\B存在一个各向同性m平面的正测度集，对于该集，每个此类平面的正交补码的平移x在一组维度dim a–m上与a相交。然后将这些结果应用于获得第n个海森堡群的类似结果。

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Intersections of Projections and Slicing Theorems for the Isotropic Grassmannian and the Heisenberg group

Abstract This paper studies the Hausdorff dimension of the intersection of isotropic projections of subsets of ℝ2n, as well as dimension of intersections of sets with isotropic planes. It is shown that if A and B are Borel subsets of ℝ2n of dimension greater than m, then for a positive measure set of isotropic m-planes, the intersection of the images of A and B under orthogonal projections onto these planes have positive Hausdorff m-measure. In addition, if A is a measurable set of Hausdorff dimension greater than m, then there is a set B ⊂ ℝ2n with dim B ⩽ m such that for all x ∈ ℝ2n\B there is a positive measure set of isotropic m-planes for which the translate by x of the orthogonal complement of each such plane, intersects A on a set of dimension dim A – m. These results are then applied to obtain analogous results on the nth Heisenberg group.

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来源期刊

Analysis and Geometry in Metric Spaces Mathematics-Geometry and Topology

CiteScore

1.80

自引率

0.00%

发文量

审稿时长

16 weeks

期刊介绍： Analysis and Geometry in Metric Spaces is an open access electronic journal that publishes cutting-edge research on analytical and geometrical problems in metric spaces and applications. We strive to present a forum where all aspects of these problems can be discussed. AGMS is devoted to the publication of results on these and related topics: Geometric inequalities in metric spaces, Geometric measure theory and variational problems in metric spaces, Analytic and geometric problems in metric measure spaces, probability spaces, and manifolds with density, Analytic and geometric problems in sub-riemannian manifolds, Carnot groups, and pseudo-hermitian manifolds. Geometric control theory, Curvature in metric and length spaces, Geometric group theory, Harmonic Analysis. Potential theory, Mass transportation problems, Quasiconformal and quasiregular mappings. Quasiconformal geometry, PDEs associated to analytic and geometric problems in metric spaces.