Markov's inequality and the existence of an extension operator for C∞ functions

Abstract

It is shown that if E is a C ∞ determining compact set in R n , then Markov's inequality for derivatives of polynomials holds on E iff there exists a continuous linear extension operator L: C ∞( E)→ C ∞( R n). Other equivalent statements (e.g., Bernstein's approximation theorem for C ∞ functions, topological linear embedding of C ∞( E) into the space of rapidly decreasing sequences of real numbers) are also given. As an application, we prove that each of those properties (of the set E) is invariant under regular analytic mappings.