I'm reading a paper "The Inhomogeneous Dirichlet Problem in Lipschitz domains" written by David Jerison and Carlos E. Kenig.
Let $u$ be a harmonic function in $\Omega$. In the page 183, the author states an inequality without proof. $$ \int_Q |\delta \nabla u|^p dx \leq \int_{Q^{*}} |u|^p dx.$$ Here $\delta(x)=\mathrm{dist} (x,\partial \Omega)$ and $Q$ is a Whitney cube for $\Omega$ and $Q^*$ denotes the double of $Q$ which also lies in $\Omega$. So $\mathrm{diam}(Q) \approx \mathrm{dist} (Q,\partial\Omega)$.
When $p=2$, allowing some constants, I can prove this estimate using some standard technique when we deduce Caccioppoli inequality.
However, I have no idea when $p\neq 2$. I tried some iteration argument when we prove the Moser estimate. In this case, the problem occurs due to constant.
I tried several approach and searched several books on harmonic function, but I failed it. How can I prove this?