## Illinois Journal of Mathematics

### 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.

#### Article information

Source
Illinois J. Math., Volume 46, Number 4 (2002), 1089-1109.

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258138468

Digital Object Identifier
doi:10.1215/ijm/1258138468

Mathematical Reviews number (MathSciNet)
MR1988252

Zentralblatt MATH identifier
1040.31004

#### Citation

Kurdyka, K.; Xiao, J. John functions, quadratic integral forms and o-minimal structures. Illinois J. Math. 46 (2002), no. 4, 1089--1109. doi:10.1215/ijm/1258138468. https://projecteuclid.org/euclid.ijm/1258138468