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 \(R^+\) by \(δ(x,y) = inf\{|I| I is a dyadic interval containing x and y\}\). Even if these functions of \(δ\) are not equivalent to \(δ\), the families of balls are the same, to wit, the dyadic intervals.