Haar wavelet characterization of dyadic Lipschitz regularity

We obtain a necessary and sufficient condition on the Haar coefficients of a real function $f$ defined on ${\mathbb{R}}^{+}$ for the Lipschitz $\alpha$ regularity of $f$ with respect to the ultrametric $\delta (x,y)=\inf \{\mathrm{\mid }I\mathrm{\mid }:x,y\in I;I\in \mathcal{D}\}$, where $\mathcal{D}$ is the family of all dyadic intervals in ${\mathbb{R}}^{+}$ and $\alpha$ is positive. Precisely, $f\in {\text{Lip}}_{\delta }(\alpha )$ if and only if $\mid <f,h_k^j>\mid \le C{2}^{-(\alpha +\frac{1}{2})j}$, for some constant $C$, every $j\in \mathbb{Z}$ and every $k=0,1,2,\dots$ Here, as usual $h_k^j(x)=2^{j\mathrm{/}2}h(2^{j}x-k)$ and $h(x)={\mathcal{X}}_{[0,1\mathrm{/}2)}(x)-{\mathcal{X}}_{[1\mathrm{/}2,1)}(x)$.