Equivalence between the growth of ∫B(x,r) ¦▽u¦p dy and T in the equation P[u] = T

We introduce a new space of Morrey type in W − 1, q (Ω) denoted M λ −1, q (Ω) . Then, we prove the following optimal condition: if u is a local solution of for some λ > N − p, 1 p + 1 q = 1 if and only if for all subsets Ω′ relatively compact, there exist C > 0, and σ > N − p such that for all x in Ω′, all R > 0: B(x, 2R) ⊂ Ω′ , we have (H) ∫ B(x,R ¦▽u¦ p dy ⩽ C · R σ . In particular, this result implies that any locally bounded solution of ( P 1) is locally Holder continuous, provided that T belongs to Mλ−1,q, for λ > N − p. Since we need boundness of solutions, we prove in the second section the boundness property of the Dirichlet problem associated to ( P 1) for a large class of T (including all the right hand side T considered in the literature). The method relies on some property of Radon measure in W− 1,q.