Vertical versus horizontal Sobolev spaces

Abstract

Let $\alpha \ge 0$, $1<p<\infty$, and let ${\mathbb{H}}^n$ be the Heisenberg group. Folland in 1975 showed that if $f:{\mathbb{H}}^n\to \mathbb{R}$ is a function in the horizontal Sobolev space $S_{2\alpha }^p({\mathbb{H}}^n)$, then $\phi f$ belongs to the Euclidean Sobolev space $S_{\alpha }^p({\mathbb{R}}^{2n+1})$ for any test function $\phi$. In short, $S_{2\alpha }^p({\mathbb{H}}^n)\subset S_{\alpha ,\mathrm{loc}}^p({\mathbb{R}}^{2n+1})$. We show that the localisation can be omitted if one only cares for Sobolev regularity in the vertical direction: the horizontal Sobolev space $S_{2\alpha }^p({\mathbb{H}}^n)$ is continuously contained in the vertical Sobolev space $V_{\alpha }^p({\mathbb{H}}^n)$. Our search for the sharper result was motivated by the following two applications. First, combined with a short additional argument, it implies that bounded Lipschitz functions on ${\mathbb{H}}^n$ have a $\frac{1}{2}$-order vertical derivative in $\mathrm{BMO}({\mathbb{H}}^n)$. Second, it yields a fractional order generalisation of the (non-endpoint) vertical versus horizontal Poincaré inequalities of V. Lafforgue and A. Naor.

26 pages