Lemma 2.9.1506216156055D

Proof. Let $e_ a$ be the $a$th coordinate vector and $h>0$. Taylor's formula with integral remainder gives

The $L^ p$ norm is invariant under translation. Moreover, $\omega (t)\, \mathrm d t=2h^{-2}(h-t)\, \mathrm d t$ is a probability measure on $[0,h]$, so Hölder's inequality gives $|\int _0^ hg\, \omega \, \mathrm d t|^ p\leq \int _0^ h|g|^ p\omega \, \mathrm d t$ for every measurable function $g$ on $[0,h]$. Applying this to $g(t)=\partial _ a^2v(x+te_ a)$, integrating in $x$ and using Tonelli's theorem, we obtain

Hence $h\| \partial _ av\| _ p\leq 2\| v\| _ p+(h^2/2)\| \partial _ a^2v\| _ p$. Since $|\nabla v|\leq \sum _ a|\partial _ av|$ and $|\partial _ a^2v|\leq |D^2v|$, summing over $a$ yields $\| \nabla v\| _ p\leq 2dh^{-1}\| v\| _ p+(dh/2)\| D^2v\| _ p$. The choice $h=2\varepsilon /d$ proves the lemma with $C_{d,\varepsilon }=d^2/\varepsilon $. $\square$