I'm going through a proof of Schoen-Simon-Yau's $L^p$ bound on the norm squared of the 2nd f.f., $|A|^2$, for stable (orientable) minimal hypersurfaces in $\Sigma \subset \mathbb{R}^n$, from Minicozzi and Colding's notes (If you have access to their course on minimal surfaces, this is theorem 2.21). The bulk of the proof is very clear but one 'lemma' is glossed over, and I'm having a hard time justifying it at all.
Namely, implicit in the proof is the truth of the following statement: If $\phi$ is Lipschitz on $\Sigma$ and of compact support, then $\phi(|A|\phi)^{1+q}$, for any $q \in [0, \sqrt{\frac{2}{n-1}})$, is also Lipschitz on $\Sigma$ (and obviously of compact support).
In fact this seems remarkable to me, and not trivial at all, since products do not preserve Lipschitz continuity in general. I've tried using local bounds on the norm squared of the 2nd f.f. but I'm just getting nowhere.
As far as I see we either need this function to be Lipschitz so that we can insert it into the stability inequality, or there is some hairier property of this function which is weaker than Lipschitz continuity but still sufficient to invoke the stability inequality. I'm hoping it's the former...
Thanks for any help!
Everything are restricted to the support of $\phi$. Note that
$|A|, \phi$ are bounded functions. (as $\Sigma$ is smooth)
$|A|$ is Lipschitz as $|\nabla|A||\le |\nabla A|$ and $|\nabla A|$ is bounded.
$|A|^{1+q}$ is Lipschitz, as it is a composition of Lipschitz functions ($f(y) = y^{1+q}$ is Lipschitz when $q\ge 0$). So is $\phi^{1+q}$.
Products of bounded Lipschitz functions are Lipschitz.
So $\phi (|A| \phi)^{1+q}$ is Lipschitz.