The theory of Besov and Triebel-Lizorkin spaces usually proceeds by taking a dyadic decomposition of unity, i.e. some non-negative functions $\psi_0,\psi \in C_c^\infty(\mathbb{R})$ such that \begin{equation*} \psi_0(x) + \sum_{n=1}^\infty \psi(2^{-n}x) = 1. \end{equation*}

My question is whether or not one may define equivalent Besov/Triebel-Lizrokin norms using a different ('logarithmic') decomposition of unity such as say \begin{equation*} \psi_0(x) + \sum_{n=1}^\infty \psi(3^{-n}x) = 1. \end{equation*}

Giving explicit definitions, one usually defines the Littlewood-Paley blocks of a function $f \in C_c^\infty(\mathbb{R})$ (taking it to be smooth/compactly supported only for illustration) via the fourier transform $\hat{f}$ \begin{gather*} \hat{f_0}(\xi) = \psi_0(\xi)\hat{f}(\xi), \\ \hat{f_n}(\xi) = \psi(2^{-n}\xi) \hat{f}(\xi), \end{gather*} or using the shorthand for Fourier multipliers, $f_0=\psi_0(D)f$, and $f_n = \psi(2^{-n}D)f$.

Then the Besov, and Triebel-Lizorkin norms ($s=0$ for illustration) are defined by \begin{gather*} \left\|f\right\|_{B^0_{p,q}} = \left(\sum_{n=0}^\infty \left\|f_n\right\|_{L^p}^q\right)^{\frac{1}{q}}, \\ \left\|f\right\|_{F^0_{p,q}} = \left\|\left(\sum_{n=0}^\infty |f_n(x)|^q\right)^{\frac{1}{q}}\right\|_{L^p}. \end{gather*}

I am aware of the Book by Triebel (Theory of Function Spaces II) which gives many equivalent characterisation of these norms. But missing in this book is the question of when instead of a dyadic decomposition of unity, I use some other ('logarithmic') decomposition of unity, such as say

\begin{equation*} \psi_0(x) + \sum_{n=1}^\infty \psi(3^{-n}x) = 1. \end{equation*}

If one then defines $f_n = \psi(3^{-n}D)f$, this likely still produces an equivalent norm, but does anyone know a reference for such a statement?