# fa.functional analysis – How to prove the reverse Hölder inequality for Laplace equations?

Let $$uin H^1(2B)$$ be a weak solution of $$Delta u=0$$ in $$2B$$, where $$B=B(0,1)$$ is a ball with center $$0$$ and radius $$1$$. Then there exists some $$p>2$$ such that
$$begin{eqnarray} left(frac{1}{|B|}int_{B}|triangledown u|^p dxright)^{1/p}leq Cleft(frac{1}{|2B|}int_{2B}|triangledown u|^2 dxright)^{1/2}. end{eqnarray}$$
where $$C$$ is an absolute constant.

I recently saw this problem and I want to get the solution of this problem. However, I meet with some troubles in it. Here is my try. First as $$u-frac{1}{|2B|}int_{2B}u$$ is also a weak solution for the Laplace equation, then by using integration by parts on the function, I can obtain that
$$begin{eqnarray} frac{1}{|B|}int_{B}|triangledown u|^2 dxleq Cleft{int_{2B}left|u-frac{1}{|2B|}int_{2B}uright|^2dxright}. end{eqnarray}$$
where $$C$$ is an absolute constant. Then, by using the Sobolev-Poincaré inequality I have
$$begin{eqnarray} left(frac{1}{|B|}int_{B}|triangledown u|^2 dxright)^{1/2}leq Cleft(frac{1}{|2B|}int_{2B}|triangledown u|^q dxright)^{1/q} end{eqnarray}$$
where $$q=frac{2d}{d+2}$$. I think it is quite similar to the final result. But I cannot go further. Can you give me some hints or references?