scholarly journals Semmes surfaces and intrinsic Lipschitz graphs in the Heisenberg group

2020 ◽  
Vol 373 (8) ◽  
pp. 5957-5996
Author(s):  
Katrin Fässler ◽  
Tuomas Orponen ◽  
Séverine Rigot
Keyword(s):  
Heisenberg Group ◽  
Lipschitz Graphs

scholarly journals Intrinsic Lipschitz Graphs and Vertical β-Numbers in the Heisenberg Group

2019 ◽  
Vol 141 (4) ◽  
pp. 1087-1147
Author(s):  
Vasileios Chousionis ◽  
Katrin Fässler ◽  
Tuomas Orponen
Keyword(s):  
Heisenberg Group ◽  
Lipschitz Graphs

scholarly journals Extensions and corona decompositions of low-dimensional intrinsic Lipschitz graphs in Heisenberg groups

2021 ◽  
Author(s):  
Daniela Di Donato ◽  
Katrin Fässler
Keyword(s):  
Heisenberg Group ◽  
Heisenberg Groups ◽  
Lipschitz Constants ◽  
Lipschitz Graphs ◽  
Low Dimensional ◽  
Lipschitz Graph

AbstractThis note concerns low-dimensional intrinsic Lipschitz graphs, in the sense of Franchi, Serapioni, and Serra Cassano, in the Heisenberg group $${\mathbb {H}}^n$$ H n , $$n\in {\mathbb {N}}$$ n ∈ N . For $$1\leqslant k\leqslant n$$ 1 ⩽ k ⩽ n , we show that every intrinsic L-Lipschitz graph over a subset of a k-dimensional horizontal subgroup $${\mathbb {V}}$$ V of $${\mathbb {H}}^n$$ H n can be extended to an intrinsic $$L'$$ L ′ -Lipschitz graph over the entire subgroup $${\mathbb {V}}$$ V , where $$L'$$ L ′ depends only on L, k, and n. We further prove that 1-dimensional intrinsic 1-Lipschitz graphs in $${\mathbb {H}}^n$$ H n , $$n\in {\mathbb {N}}$$ n ∈ N , admit corona decompositions by intrinsic Lipschitz graphs with smaller Lipschitz constants. This complements results that were known previously only in the first Heisenberg group $${\mathbb {H}}^1$$ H 1 . The main difference to this case arises from the fact that for $$1\leqslant k<n$$ 1 ⩽ k < n , the complementary vertical subgroups of k-dimensional horizontal subgroups in $${\mathbb {H}}^n$$ H n are not commutative.

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

The horofunction boundary of the Heisenberg group

2009 ◽  
Vol 242 (2) ◽  
pp. 299-310 ◽  
Author(s):  
Tom Klein ◽  
Andrew Nicas
Keyword(s):  
Heisenberg Group

scholarly journals Weighted CBMO estimates for commutators of matrix Hausdorff operator on the Heisenberg group

2020 ◽  
Vol 18 (1) ◽  
pp. 496-511
Author(s):  
Amna Ajaib ◽  
Amjad Hussain
Keyword(s):  
Heisenberg Group ◽  
Herz Spaces ◽  
Hausdorff Operator ◽  
Hausdorff Operators

Abstract In this article, we study the commutators of Hausdorff operators and establish their boundedness on the weighted Herz spaces in the setting of the Heisenberg group.

scholarly journals Plenty of big projections imply big pieces of Lipschitz graphs

2021 ◽  
Author(s):  
Tuomas Orponen
Keyword(s):  
Lipschitz Graphs ◽  
Regular Sets ◽  
Big Pieces

AbstractI prove that closed n-regular sets $$E \subset {\mathbb {R}}^{d}$$ E ⊂ R d with plenty of big projections have big pieces of Lipschitz graphs. In particular, these sets are uniformly n-rectifiable. This answers a question of David and Semmes from 1993.

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