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

María Florencia Acosta, Hugo Aimar, Ivana Gómez, Federico Morana

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

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 , the diffusion metric is a function of the dyadic distance, given in by . Even if these functions of are not equivalent to , the families of balls are the same, to wit, the dyadic intervals.
Paper Structure (3 sections, 4 theorems, 23 equations)

This paper contains 3 sections, 4 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. Main results

Key Result

Proposition 1.1

Let $d_t$ be defined as before. Then

Theorems & Definitions (10)

  • Proposition 1.1
  • proof
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof