Haar wavelet characterization of dyadic Lipschitz regularity

Abstract

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

6 pages. This manuscript has been accepted for publication in the Revista de la Unión Matemática Argentina