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Let  u  be a weak solution of the equation (-Δ)  m   u  =  f  with Dirichlet boundary conditions in a smooth bounded domain Ω ⊂ ℝ  n  . In this paper, we obtain some estimates for the Green's function associated to this problem. Moreover, under appropriate conditions on  p , we prove some weighted Sobolev a priori estimates for the solution  u , where the weight is a power of the distance function. This result extends previous work by R. G. Durán and M. Sanmartino ([Indiana Univ. Math. J. 57 (2008), no. 7, 3463–3478] and [Anal. Theory Appl. 26 (2010), no. 4, 339–349]).

Weighted a priori estimates with powers of the distance function for elliptic equations