Effective quasianalytic Remez inequalities on tame sets

Abstract

We establish a Remez inequality for functions in quasianalytic Denjoy-Carleman classes ${\mathcal{C}}_M$ on a large family of fat compact sets $K\subseteq {\mathbb{R}}^n$ with tame geometry, including all fat compact sets definable in the o-minimal expansion of $\mathbb{R}$ by restricted ${\mathcal{C}}_M$ functions. The inequality generalizes the classical Remez inequality for polynomials and its quasianalytic versions on convex bodies, replacing the polynomial degree by the Bang degree, an integer associated with the weight $M$ and the size of the function. The constants depend explicitly on the Bang degree and the geometry of $K$. We derive a range of quantitative consequences: estimates for the volume of sublevel sets, comparison of $L^p$-norms, effective inequalities of Lojasiewicz, Harnack, and Markov type with explicit constants, and decay estimates for oscillatory integrals.

50 pages