垂直与水平索伯列夫空间
Vertical versus horizontal Sobolev spaces
摘要：
设$α\geq 0$、$1 < p < \infty$和$\mathbb{H}^{n}$为海森堡群。Folland在1975年表明，如果$f \colon \mathbb{H}^{n} \to \mathbb{R}$是一个功能，在水平索伯列夫空间$S^{p}_{2α}(\mathbb{H}^{n})$，然后$φf$属于欧几里德索伯列夫空间$S^{p}_α(\mathbb{R}^{2n + 1})$的任何测试功能$φ$。简而言之，就是$S^{p}_{2α}(\mathbb{H}^{n}) \subset S^{p}_{α,\mathrm{loc}}(\mathbb{R}^{2n + 1})$。我们表明，本地化可以省略，如果一个只关心在垂直方向上的Sobolev正则性：水平Sobolev空间$S_{2α}^{p}(\mathbb{H}^{n})$是连续包含在垂直Sobolev空间$V^{p}_α(\mathbb{H}^{n})$。我们对更清晰结果的搜索是由以下两个应用引起的。首先，结合一个简短的额外的参数，它意味着有界Lipschitz函数$\mathbb{H}^{n}$有$\tfrac{1}{2}$阶垂直导数在$\mathrm{BMO}(\mathbb{H}^{n})$。其次，它给出了V.Lafforgue和A. Naor.
英文摘要：
Let $α\geq 0$, $1 < p < \infty$, and let $\mathbb{H}^{n}$ be the Heisenberg group. Folland in 1975 showed that if $f \colon \mathbb{H}^{n} \to \mathbb{R}$ is a function in the horizontal Sobolev space $S^{p}_{2α}(\mathbb{H}^{n})$, then $φf$ belongs to the Euclidean Sobolev space $S^{p}_α(\mathbb{R}^{2n + 1})$ for any test function $φ$. In short, $S^{p}_{2α}(\mathbb{H}^{n}) \subset S^{p}_{α,\mathrm{loc}}(\mathbb{R}^{2n + 1})$. We show that the localisation can be omitted if one only cares for Sobolev regularity in the vertical direction: the horizontal Sobolev space $S_{2α}^{p}(\mathbb{H}^{n})$ is continuously contained in the vertical Sobolev space $V^{p}_α(\mathbb{H}^{n})$. 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 $\tfrac{1}{2}$-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.
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