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

Abstract

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 $t \gt 0$, the diffusion metric is a function of the dyadic distance, given in $\mathbb{R}^{+}$ by $\delta(x,y) = \inf\{|I|\colon I \text{ is a dyadic interval containing } x \text{ and } y\}$. Even if these functions of $\delta$ are not equivalent to $\delta$, the families of balls are the same, to wit, the dyadic intervals.