Growth and decay of H\"older moduli

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.