Let $\mathbb{H}$ be the first Heisenberg group, and let $k\in C^{\infty }(\mathbb{H}\setminus \{0\})$ be a kernel which is either odd or horizontally odd, and satisfies$|{\mathrm{\nabla }}_{\mathbb{H}}^{n}k(p)|\le C_n{\Vert p\Vert }^{-1-n},p\in \mathbb{H}\setminus \{0\},n\ge 0.$ The simplest examples include certain Riesz-type kernels first considered by Chousionis and Mattila, and the horizontally odd kernel $k(p)={\mathrm{\nabla }}_{\mathbb{H}}\log \Vert p\Vert$. We prove that convolution with k, as above, yields an $L^2$-bounded operator on regular curves in $\mathbb{H}$. This extends a theorem of G. David to the Heisenberg group.

As a corollary of our main result, we infer that all 3-dimensional horizontally odd kernels yield $L^2$ bounded operators on Lipschitz flags in $\mathbb{H}$. This is needed for solving sub-elliptic boundary value problems on domains bounded by Lipschitz flags via the method of layer potentials. The details are contained in a separate paper. Finally, our technique yields new results on certain non-negative kernels, introduced by Chousionis and Li.

让 $\mathbb{H}$ 成为第一个海森堡群，让 $克\in C^{\infty }(\mathbb{H}\setminus \{0\})$ 是奇数或水平奇数的核，并且满足$|{\mathrm{\nabla }}_{\mathbb{H}}^n克(磷)|\le C_n{\Vert 磷\Vert }^{-1-n},磷\in \mathbb{H}\setminus \{0\},n\ge 0.$ 最简单的例子包括 Chousionis 和 Mattila 首先考虑的某些 Riesz 型核，以及水平奇数核 $克(磷)={\mathrm{\nabla }}_{\mathbb{H}}\mathrm{日志}\Vert 磷\Vert$. 我们证明了与k 的卷积，如上所述，产生一个${升}^2$正则曲线上的有界算子 $\mathbb{H}$. 这将 G. David 的定理扩展到海森堡群。

作为我们主要结果的推论，我们推断所有 3 维水平奇数核产生 ${升}^2$上界运营商李氏标志在$\mathbb{H}$. 这是通过层势方法解决以 Lipschitz 标志为界的域上的亚椭圆边界值问题所需要的。详细信息包含在单独的论文中。最后，我们的技术在 Chousionis 和 Li 介绍的某些非负核上产生了新的结果。