Łojasiewicz inequality at singular points

Abstract:

We prove a generalized version of Łojasiewicz’s inequality. The famous Łojasiewicz inequality asserts that if $f$ is a ${\mathcal C}^1$ globally subanalytic function in a neighborhood of a point $a \in \mathbb {R}^n$, there is a neighborhood $U$ of $a$ and a rational number $\theta \in [0,1)$ as well as a constant $C$ such that $|f(x)-f(a)|^\theta \le C |\nabla _x f|$ for $x\in U$ (where $\nabla _x f$ stands for the gradient of the function $f$ at $x$). We give an inequality of the same type that applies to the case where $a$ is not an interior point of the domain of $f$.