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

Abstract

This note concerns low-dimensional intrinsic Lipschitz graphs, in the sense of Franchi, Serapioni, and Serra Cassano, in the Heisenberg group {mathbb {H}}^n, nin {mathbb {N}}. For 1leqslant kleqslant n, we show that every intrinsic L-Lipschitz graph over a subset of a k-dimensional horizontal subgroup {mathbb {V}} of {mathbb {H}}^n can be extended to an intrinsic L'-Lipschitz graph over the entire subgroup {mathbb {V}}, where L' depends only on L, k, and n. We further prove that 1-dimensional intrinsic 1-Lipschitz graphs in {mathbb {H}}^n, nin {mathbb {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. The main difference to this case arises from the fact that for 1leqslant k<n, the complementary vertical subgroups of k-dimensional horizontal subgroups in {mathbb {H}}^n are not commutative.