John functions, quadratic integral forms and o-minimal structures

Abstract

Let $\Omega$ be a proper subdomain of $\mathbb{R}^n$, $n\ge 2$, and let $\partial{\Omega}$ and $\delta_{\Omega}(x)$ denote, respectively, the boundary of $\Omega$ and the Euclidean distance of the point $x\in \Omega$ to $\mathbb{R}^n \setminus\Omega$. Denote by $K(\Omega)$ the John space of all $C^1$ functions $f:\Omega\rightarrow\mathbb{R}$ with $\sup_{x\in \Omega}\delta_\Omega (x)|\nabla f(x)|<+\infty$. We study $K(\Omega)$-functions via quadratic integral forms and o-minimal structures.