Approximation of critical regularity functions on stratified homogeneous groups

Eduard Valentin Curca (1)

1 ICJ - Institut Camille Jordan

Let $G$ be a stratified homogeneous group with homogeneous dimension $Q$ and whose Lie algebra is generated by the left-invariant vector fields $X_{1}$,...,$X_{d_{1}}$. Let $p,q\in(1,\infty)$, $\alpha=Q/p$ and $\delta>0$. We prove that for any function $f\in\dot{F} _{q}^{\alpha ,p}(G)$ there exists a function $F\in{L}^{\infty}(G)\cap\dot{F} _{q}^{\alpha,p}(G)$ such that \begin{eqnarray*} \sum_{i=1}^{\mathbf{k}}\left\Vert X_{i}(f-F)\right\Vert _{\dot{F}% _{q}^{\alpha -1,p}(G)} &\leq &\delta \left\Vert f\right\Vert _{\dot{F}% _{q}^{\alpha ,p}(G)}\text{,} \\ \left\Vert F\right\Vert _{L^{\infty }(G)}+\left\Vert F\right\Vert _{\dot{F}% _{q}^{\alpha ,p}(G)} &\leq &C_{\delta }\left\Vert f\right\Vert _{\dot{F}% _{q}^{\alpha ,p}(G)} \end{eqnarray*} where $\mathbf{k}$ is the largest integer smaller than $min(p,d_{1})$ and $ C_{\delta } $ is a positive constant only depending on $\delta$. Here, $\dot{ F}_{q}^{\alpha,p}(G)$ is a Triebel-Lizorkin type space adapted to $G$. \\ 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.

Mots clés: stratified homogeneous groups; critical Triebel–Lizorkin spaces

Domaines: Mathématiques [math]; Equations aux dérivées partielles [math.AP]

Eduard Valentin Curca. Approximation of critical regularity functions on stratified homogeneous groups. Communications in Contemporary Mathematics, 2021, 23 (6), pp.2050059. ⟨10.1142/S0219199720500595⟩. ⟨hal-02305322v2⟩