Valentina Franceschi, Roberto Monti, Alessandro Socionovo
We prove (sub)mean value formulas at the point $0\inΣ$ for (sub)harmonic functions a on a hypersurface $Σ\subset\mathbb{R}^{n+1}$ where the differentiable structure and the surface measure depend on the ambient Grushin structure.
This paper contains 8 sections, 10 theorems, 96 equations, 1 figure.
Theorem 1.1
Let $\Sigma\subset\mathbb{R}^{n+1}$ be an $\alpha$-harmonic hypersurface of class $C^2$ with $0\in\Sigma$. Any function $f \in C^2(\Sigma)$ such that $\mathcal{L}_\Sigma f=0$ satisfies the mean-value formula at $0$ for all $r\in(0,r_0)$ and for some $r_0>0$ depending on $\Sigma$. The constant $0<C_{\Sigma,n,\alpha}<\infty$ is defined by where the right hand-side does not depend on $r\in (0,r_0)$.
