Growth and decay of Hölder moduli

If $f:{\mathbf{R}}^d\to \mathbf{C}$ is bounded and $f$’s Hölder $\alpha$-modulus of continuity grows no faster than $(1+\mathrm{\mid }x\mathrm{\mid }{)}^M$ ($M\ge 0$) then, for every $\epsilon >0$, there is a $\beta >0$ such that $f$’s Hölder $\beta$-modulus grows no faster than $(1+\mathrm{\mid }x\mathrm{\mid }{)}^{\epsilon }$. We use this easy fact to show that, if $\mathrm{\mid }f\mathrm{\mid }$ decays as fast as $(1+\mathrm{\mid }x\mathrm{\mid }{)}^{-R}$ (for $R>0$) and $f$’s $\alpha$-Hölder modulus grows no faster than $(1+\mathrm{\mid }x\mathrm{\mid }{)}^M$, then, for every $0\le R^{\mathrm{\prime }}<R$, there is a $\beta >0$ such that $f$’s $\beta$-Hölder modulus decays as fast as $(1+\mathrm{\mid }x\mathrm{\mid }{)}^{-R^{\mathrm{\prime }}}$. 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.