Wavelet transforms for homogeneous mixed-norm Triebel--Lizorkin spaces

Abstract

Homogeneous mixed-norm Triebel--Lizorkin spaces are introduced and studied with the use of a discrete wavelet transformation, the so-called $φ$-transform. This extends the classical $φ$-transform approach introduced by Frazier and Jawerth to the setting of mixed-norm spaces. Moreover, the theory of the $φ$-transform is enhanced through a precise definition of the synthesis operator, in terms of a Pettis integral, and a number of rigorous results for this operator. Especially its terms can always be summed in any order, without changing the resulting distribution.

25 pages. Link added. Revised version. Published online 25 March 2017 in Monatshefte für Mathematik. Final version available at http://www.doi.dx/10.1007/s00605-017-1036-z