Lower bounds for quasianalytic functions, I. How to control smooth functions

Abstract

Let $\mathscr{F}$ be a class of functions with the uniqueness property: if $f\in \mathscr{F}$ vanishes on a set $E$ of positive measure, then $f$ is the zero function. In many instances, we would like to have a quantitative version of this property, e.g. the estimate from below for the norm of the restriction operator $f\mapsto f\big|_E$ or, equivalently, a lower bound for $|f|$ outside a small exceptional set. Such estimates are well-known and useful for polynomials, complex- and real-analytic functions, exponential polynomials. In this work we prove similar results for the Denjoy-Carleman and the Bernstein classes of quasianalytic functions. In the first part, we consider quasianalytically smooth functions. This part relies upon Bang's approach and includes the proofs of relevant results of Bang. In the second part, which is to be published separately, we deal with classes of functions characterized by exponentially fast approximation by polynomials whose degrees belong to a given very lacunar sequence.The proofs are based on the elementary calculus technique.