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 > 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.