Łojasiewicz inequalities in o-minimal structures

Abstract

This note is devoted to the generalization of Łojasiewicz inequalities for functions definable in o-minimal structures, which is, roughly speaking, a generalization for semialgebraic or global subanalytic functions. We present some o-minimal versions of the inequalities to compare two definable functions globally or in some neighborhoods of the zero-sets of the functions, and the gradient inequalities (Kurdyka–Łojasiewicz inequality and Bochnak–Łojasiewicz inequality). Some applications of the inequalities are given.