Wavelet transforms for homogeneous mixed-norm Triebel-Lizorkin spaces

Homogeneous mixed-norm Triebel-Lizorkin spaces are introduced and studied with the use of a discrete wavelet transformation, the so-called phi-transform. This extends the classical phi-transform approach introduced by Frazier and Jawerth to the setting of mixed-norm spaces. Moreover, the theory of the phi-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.