On the structure of the diffusion distance induced by the fractional dyadic Laplacian

In this note we explore the structure of the diffusion metric of Coifman-Lafon determined by fractional dyadic Laplacians. The main result is that, for each \({t>0}\), the diffusion metric is a function of the dyadic distance, given in \(\mathbb{R}^+\) by \(\delta(x,y) = \inf\{|I|: I \text{ is a dyadic interval containing } x \text{ and } y\}\). Even if these functions of \(\delta\) are not equivalent to \(\delta\), the families of balls are the same, to wit, the dyadic intervals.