Approximation of critical regularity functions on stratified homogeneous groups

Abstract

Let [Formula: see text] be a stratified homogeneous group with homogeneous dimension [Formula: see text] and whose Lie algebra is generated by the left-invariant vector fields [Formula: see text]. Let [Formula: see text], [Formula: see text] and [Formula: see text]. We prove that for any function [Formula: see text] there exists a function [Formula: see text] such that [Formula: see text] [Formula: see text] where [Formula: see text] is the largest integer smaller than [Formula: see text] and [Formula: see text] is a positive constant depending only on [Formula: see text]. Here, [Formula: see text] is a homogeneous Triebel–Lizorkin type space adapted to [Formula: see text]. This generalizes earlier results of Bourgain, Brezis [New estimates for eliptic equations and Hodge type systems, J. Eur. Math. Soc. 9(2) (2007) 277–315] and of Bousquet, Russ, Wang, Yung [Approximation in fractional Sobolev spaces and Hodge systems, J. Funct. Anal. 276(5) (2019) 1430–1478] in the Euclidean case and answers an open problem in the latter reference.