scholarly journals Sharp estimates of the geometric rigidity on the first Heisenberg group

2019 ◽  
Vol 488 (6) ◽  
pp. 590-594
Author(s):  
D. V. Isangulova
Keyword(s):  
Heisenberg Group ◽  
Riemannian Geometry ◽  
Uniform Norm ◽  
Sobolev Norm ◽  
John Domain ◽  
Quantitative Stability ◽  
Sharp Estimates ◽  
Geometric Rigidity

We prove quantitative stability of isometries on the first Heisenberg group with sub-Riemannian geometry: every (1 + )-quasi-isometry of the John domain of the Heisenberg group is close to some isometry with order of closeness in the uniform norm and with the order of closeness+ in the Sobolev norm. An example demonstrating the asymptotic sharpness of the results is given.

scholarly journals Quantitative Stability in the Geometry of Semi-discrete Optimal Transport

2020 ◽  
Author(s):  
Mohit Bansil ◽  
Jun Kitagawa
Keyword(s):  
Optimal Transport ◽  
Convex Geometry ◽  
Regularity Theory ◽  
Uniform Norm ◽  
Quantitative Stability ◽  
Wirtinger Inequality ◽  
Dual Variables ◽  
Transport Problems ◽  
Monge Ampere ◽  
Stability Results

Abstract We show quantitative stability results for the geometric “cells” arising in semi-discrete optimal transport problems. We first show stability of the associated Laguerre cells in measure, without any connectedness or regularity assumptions on the source measure. Next we show quantitative invertibility of the map taking dual variables to the measures of Laguerre cells, under a Poincarè-Wirtinger inequality. Combined with a regularity assumption equivalent to the Ma–Trudinger–Wang conditions of regularity in Monge-Ampère, this invertibility leads to stability of Laguerre cells in Hausdorff measure and also stability in the uniform norm of the dual potential functions, all stability results come with explicit quantitative bounds. Our methods utilize a combination of graph theory, convex geometry, and Monge-Ampère regularity theory.

Subsolutions That Are Close in the Uniform Norm Are Close in the Sobolev Norm as Well

2012 ◽  
Vol 66 (1) ◽  
pp. 1-25
Author(s):  
Muhammad Shoaib Saleem ◽  
Malkhaz Shashiashvili
Keyword(s):  
Uniform Norm ◽  
Sobolev Norm

Weighted estimates for commutators of Hausdorff operators on the Heisenberg group

2020 ◽  
pp. 39-62
Author(s):  
Nguyen Minh Chuong ◽  
◽  
Dao Van Duong ◽  
Nguyen Duc Duyet ◽  
◽  
...  
Keyword(s):  
Heisenberg Group ◽  
Weighted Estimates ◽  
Hausdorff Operators
Sign in / Sign up

Export Citation Format

Share Document