On Dilation Operators in Besov Spaces

Resumo

We consider dilation operators Tk : f f (2k ·) in the framework of Besov spaces Bp,q (Rn ) when 0 < p 1. If s > n p - 1 , Tk is a bounded linear s 1 operator from Bp,q (Rn ) into itself and there are optimal bounds for its norm. s 1 We study the situation on the line s = n 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 diver sity of different approaches to Besov spaces on this line, associated to definitions by differences, Fourier-analytical methods, and subatomic decompositions.