Let $\Omega\subset\mathbb{R}^N$ be a bounded open set and $X=(X_1,X_2,\ldots,X_k)$ be a smooth vector field satisfying the Hormanders finite rank (of rank $N$) condition. Suppose $1<p<\infty$ and let $W_0^{1,p}(\Omega)$ is the Sobolev space defined as the closure of $C_c^{\infty}(\Omega)$ under the norm $$ \|u\|=\|Xu\|_{L^p(\Omega)}, $$ where $Xu=(X_1 u,X_2 u,\ldots,X_k u)$ is the gradient of $u:\Omega\to\mathbb{R}$. Can someone please inform if the space $W_0^{1,p}(\Omega)$ is a uniform convex space? It is known that $W_0^{1,p}(\Omega)$ is a separable and reflexive Banach space, but unable to find results concerning the uniform convexity. Any reference would also be highly appreciated.
Thank you very much.