On Dilation Operators in Besov Spaces

Abstract

We consider dilation operators T_k : f → f(2^k.) in the framework of Besov spaces B^s_p,q (Rⁿ) when 0 < p≤ 1. If s > n(1/p — 1) Tk is a bounded linear operator from B^s_p,q (Rⁿ) into itself and there are optimal bounds for its norm. We study the situation on the line s = n(1/p — 1), an open problem mentioned in [5, 2.3.1, 2.3.2]. It turns out that the results shed new light upon the diversity of different approaches to Besov spaces on this line, associated to definitions by differences, Fourier-analytical methods, and subatomic decompositions.