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Pré-Publication, Document De Travail Année : 2019

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].
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Dates et versions

hal-02305322 , version 1 (04-10-2019)
hal-02305322 , version 2 (05-02-2020)

Identifiants

  • HAL Id : hal-02305322 , version 1

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Eduard Curcă. Approximation of critical regularity functions on stratified homogeneous groups. 2019. ⟨hal-02305322v1⟩
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