Approximation of critical regularity functions on stratified homogeneous groups
Abstract
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.
Origin : Files produced by the author(s)
Comment : Ce pdf est la version preprint de l'article (version soumise à l'éditeur, avant peer-reviewing)
Comment : Ce pdf est la version preprint de l'article (version soumise à l'éditeur, avant peer-reviewing)
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