ON DILATION OPERATORS IN TRIEBEL-LIZORKIN SPACES

We consider dilation operators T-k : f -> f (2(k).) in the framework of Triebel-Lizorkin spaces F-p,q(s) (R-n). If s > n max (1/p - 1, 0), T-k is a bounded linear operator from F-p,q(s) (R-n) into itself and there are optimal bounds for its norm. We study the situation on the line s = n max (1/p - 1, 0, an open problem mentioned in [ET96, 2.3.1]. It turns out that the results shed new light upon the diversity of different approaches to Triebel-Lizorkin spaces on this line, associated to definitions by differences, Fourier-analytical methods and subatomic decompositions.