Sharp Estimates for Lipschitz Class

Abstract

Let $I$ be an interval contained in ${\mathbb {R}}$. For a given function $f:I\rightarrow {\mathbb {R}}$, $u\in I$ and any $0<\alpha \le 1$, set

$$\begin{aligned} f_\alpha ^\sharp (u)=\sup \frac{1}{|J|^{\alpha }}\left( \frac{1}{|J|}\int _J \left| f(x)-\frac{1}{|J|}\int _J f(y)\text{ d }y\right| ^2\text{ d }x\right) ^{1/2}, \end{aligned}$$

where the supremum is taken over all subintervals $J\subseteq I$ which contain $u$. The paper contains the proofs of the estimates

$$\begin{aligned} \ell (\alpha )\big |\big |f_\alpha ^\sharp \big |\big |_{L^\infty (I)}\le \big |\big |f\big |\big |_{{\text {Lip}}_\alpha (I)}\le L(\alpha )\big |\big |f_\alpha ^\sharp \big |\big |_{L^\infty (I)}, \end{aligned}$$

where

$$\begin{aligned} \ell (\alpha )=2\sqrt{2\alpha +1},\quad L(\alpha )=\frac{(4\alpha +4)^{(\alpha +1)/(2\alpha +1)}\sqrt{2\alpha +1}}{2\alpha } \end{aligned}$$

are the best possible. The proof rests on the evaluation of Bellman functions associated with the above estimates.