Continuity estimates for p-Laplace type operators in Orlicz-Zygmund spaces

We study the Dirichlet problem divAðx;‘uÞ 1⁄4 div f in W; u1⁄40 onqW; in a bounded Lipschitz domain WHRN, with Nb2. The vector field A:WRN !RN satisfies the tipical growth and coercivity conditions of the p-Laplacian type operator with p > 1. We prove existence and uniqueness results in the case the vector field f belongs to the Orlicz–Zygmund space qa bNp L log LðloglogLÞ ðW;R Þ,q1⁄4p1,a>0andbaRora1⁄40andb>0.Inparticular,the gradient of the solution belongs to L p log a Lðlog log LÞ b ðW; R N Þ. Further, we provide estimates implying the continuity of the operator which carries any given f into the gradient field ‘u of the solution.