Tame linear extension operators for smooth Whitney functions

Abstract

For a compact set K ⊆ R d we present a rather easy construction of a linear extension operator E : E ( K ) → C ∞ ( R d ) for the space of Whitney jets E ( K ) which satisfies linear tame continuity estimates sup { | ∂ α E ( f ) ( x ) | : | α | ⩽ m , x ∈ R d } ⩽ C m , ε ‖ f ‖ ( r + ε ) m , where ‖ ⋅ ‖ s denotes the s-th Whitney norm. The construction turns out to be possible if and only if the local Markov inequality LMI ( s ) introduced by Bos and Milman holds for every s > r on K. In particular, E ( K ) admits a tame linear extension operator if and only if the local Markov inequality LMI ( s ) holds on K for some s ⩾ 1 .