Sharp estimates for Lipschitz class. (English) Zbl 1342.26007
Summary: Let \(I\) be an interval contained in \(\mathbb R\). For a given function \(f:I\to\mathbb R\), \(u\in I\) and any \(0<\alpha\leq 1\), set
\[
f_\alpha^\#(u)=\sup\frac{1}{|J|^\alpha}\left(\frac{1}{|J|}\int_J\left|f(x)-\frac{1}{|J|}\int_Jf(y)\text{d}y\right|^2\text{d}x\right)^{1/2},
\]
where the supremum is taken over all subintervals \(J\subseteq I\) which contain \(u\). The paper contains the proofs of the estimates
\[
\ell(\alpha)\| f_\alpha^\#\|_{L^\infty(I)}\leq \| f\|_{\text{Lip}_\alpha(I)}\leq L(\alpha)\| f_\alpha^\#\|_{L^\infty (I)},
\]
where
\[
\ell(\alpha)=2\sqrt{2\alpha +1},\quad L(\alpha)=\frac{(4\alpha +4)^{(\alpha+1)/(2\alpha+1)}\sqrt{2\alpha+1}}{2\alpha}
\]
are the best possible. The proof rests on the evaluation of Bellman functions associated with the above estimates.
MSC:
| 26A16 | Lipschitz (Hölder) classes |
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