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 $\delta (x,y)=\inf \{\mathrm{\mid }I\mathrm{\mid }:I\text{ is a dyadic interval containing }x\text{ and }y\}$. Even if these functions of $\delta$ are not equivalent to $\delta$, the families of balls are the same, to wit, the dyadic intervals.