Approximation of critical regularity functions on stratified homogeneous groups
Résumé
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
[4] and of Bousquet, Russ, Wang, Yung [6] in the Euclidean
case and answers an open problem in [6].
Origine | Fichiers produits par l'(les) auteur(s) |
---|
Loading...