On dilation operators in Triebel-Lizorkin spaces

We consider dilation operators Tk : f → f(2 k ·) in the framework of Triebel-Lizorkin spaces F s p,q(R ). If s > n max ` 1 p − 1, 0 ́ , Tk is a bounded linear operator from F s p,q(R ) 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. MSC (2000): 46E35