Growth and decay of H\"older moduli

TL;DR

This paper investigates the relationship between the growth of H"older moduli and decay rates of functions, providing results that connect these properties and applying them to improve understanding of vaguelet families.

Contribution

It establishes a link between the growth of H"older moduli and decay rates, and applies this to strengthen results on vaguelet almost-orthogonality.

Findings

H"older modulus growth can be controlled by decay rates of functions.

For any decay rate below a threshold, a corresponding H"older modulus exists.

Strengthens a result of Coifman and Meyer on vaguelet almost-orthogonality.

Abstract

If $f:{\bf R}^d\to{\bf C}$ is bounded and $f$'s H\"older $\alpha$-modulus of continuity grows no faster than $(1+\vert x\vert)^M$ ($M\geq0$) then, for every $\epsilon>0$, there is a $\beta>0$ such that $f$'s H\"older $\beta$-modulus grows no faster than $(1+\vert x\vert)^{\epsilon}$. We use this easy fact to show that, if $\vert f\vert$ decays as fast as $(1+\vert x\vert)^{-R}$ (for $R>0$) and $f$'s $\alpha$-H\"older modulus grows no faster than $(1+\vert x\vert)^M$, then, for every $0\leq R'< R$, there is a $\beta>0$ such that $f$'s $\beta$-H\"older modulus decays as fast as $(1+\vert x\vert)^{-R'}$. We apply this to strengthen a result of Coifman and Meyer on almost-orthogonality of vaguelet families and to derive other useful facts about vaguelets and vaguelet-like functions.