  Let $u$ be a weak solution of $ (-\Delta)^m u=f $ with Dirichlet boundary
conditions in a smooth bounded domain $\Omega \subset \mathbb{R}^n$.
  Then, the main goal of this paper is to prove the following a priori
estimate: $$ \|u\|_{W^{2m,p}_\omega(\Omega)} \le C\,
\|f\|_{L^p_\omega(\Omega)}, $$ where $\omega$ is a weight in the Muckenhoupt
class $A_p.$
