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Let $f$ be a non-negative $C^2$ function on a compact domain $\Omega$ in $\mathbb R^n$. I am trying to prove the inequality

$$\|\sqrt f\|_{C^{0,1}(\Omega)}\leq C(1 + \|f\|_{C^{1,1}(\Omega)})$$

where $C^{k,\alpha}(\Omega)$ denotes the Holder space. It seems like this should be a consequence of the mean value theorem and/or the fundamental theorem of calculus, but I am not seeing an elementary proof of this nature. Any suggestions?

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  • $\begingroup$ Jensen's Inequality? $\endgroup$ – treble Dec 21 '12 at 21:20
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It is incorrect. For example, let $n=1$, $\Omega=[0,1]$ and $f(x)=x$. Then $\|\sqrt{f}\|_{C^{0,1}(\Omega)}=\infty$.

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  • $\begingroup$ Does the inequality hold if $M$ is instead assumed to be a compact manifold? $\endgroup$ – user15464 Dec 21 '12 at 21:42
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    $\begingroup$ @user15464: If you mean $M$ is compact without boundary, I think so. The problem here is that the zero of $f$ is located at the boundary of $\Omega$. If $f(x)=0$ for some interior point $x$, then since $f\ge 0$, $f'(x)=0$, so $\sqrt{f}$ is well behaved around $x$. $\endgroup$ – 23rd Dec 21 '12 at 22:00
  • $\begingroup$ Can you give an outline of a proof? $\endgroup$ – user15464 Dec 21 '12 at 22:10
  • $\begingroup$ @user15464: Sorry, my last comment was misleading. I think in your inequality, $C$ is supposed to be dependent only on $\Omega$ or $M$. Then the inequality fails even when $M$ is a compact manifold without boundary. $\endgroup$ – 23rd Dec 21 '12 at 23:11

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