On Kac's principle of not feeling the boundary for the Kohn Laplacian on the Heisenberg group (Q891427)

In a nutshell, the authors provide sufficient conditions for having a unique solution for a sub-Laplacian equation on the Heisenberg group \(\mathbb H_{n-1}\). Precisely, they consider the operator \(\square_{a,b}=\sum_{l=1}^{n-1}(aX_j\overline{X}_l+b\overline{X_l}X_l)\) such that \(a+b=n-1\) and \((X_l)_{1\leq\l\leq n-1}\) represents the family of left-invariant vector fields of \(\mathbb H_{n-1}\). Then, for \(\Omega\) an open smooth bounded set in \(\mathbb H_{n-1}\), the authors state that \(\square_{a,b}(\cdot)=c_{a,b}f\) has an unique integral-solution (the Newton potential for \(f\)) defined by \(\int_\Omega f(\xi)\varepsilon(\xi^{-1}z)d\mu(\xi)=u(z)\in C^2(\Omega)\cap C^1(\overline{\Omega})\) such \(\varepsilon(\xi^{-1}z)\) is an explicit rescaled fundamental solution for \(\square_{a,b}\) on \(\mathbb H_{n-1}\), \(\mu\) is the Lebesgue measure on \(\mathbb C^{n-1}\times\mathbb R\), \(c_{a,b}\) representing an explicit complex constant relying on \(a,b\), and \(f\) is a complex-valued function on \(\Omega\) such that \(|f(z_1)-f(z_2)|\leq K|z_2^{-1}z_1|^{\alpha}\) for all distinct pairs \((z_1,z_2)\in \Omega^2\) such that \(\alpha\in (0,1)\), and \(K\) is a strictly positive constant. Furthermore, for \(z\) belonging to \(\partial \Omega\), the boundary of \(\Omega\), the solution satisfies NEWLINE\[NEWLINE\big(c_{a,b}-H.R(z)\big)u(z)-\int_{\partial\Omega}\varepsilon(\xi,z)<\nabla^{b,a}u(\xi),d\mu(\xi)>+p.v\int_{\partial\Omega}u(\xi)<\nabla^{a,b}\varepsilon(\xi,z),d\mu(z)>=0,NEWLINE\]NEWLINE where \(H.R(z)\) is the half residue furnished by \(p.v\int_{\partial\Omega}<\nabla^{a,b}\varepsilon(\xi,z),d\mu(z)>\) and \(\nabla^{a,b}(\cdot):=\sum_{l=1}^{n-1}aX_l(\cdot)\overline{X_l}+b\overline{X}_l(\cdot)X_l\) (Theorem 2.1). Among the tools used for the proof, the authors apply the generalized Green's second formula to \(c_{a,b}u(z)\). The uniqueness solution for the generalized sub-Laplacian equation \(\square_{a,b}^mu(z)=c_{a,b}f(z)\), (\(m\in\{1,2,\ldots\}\)) is stated in the third section (Theorem 3.1).