J. reine angew. Math. 584 (2005), 117—148 Journal fu ¨r die reine und angewandte Mathematik ( Walter de Gruyter Berlin  New York 2005 Gradient estimates for the pðxÞ-Laplacean system By Emilio Acerbi and Giuseppe Mingione at Parma Abstract. We prove Caldero ´n and Zygmund type estimates for a class of elliptic problems whose model is the non-homogeneous pðxÞ-Laplacean system divðjDuj pðxÞ2 DuÞ¼divðjF j pðxÞ2 F Þ: Under optimal continuity assumptions on the function pðxÞ > 1 we prove that jF j pðxÞ A L q loc )jDuj pðxÞ A L q loc Eq > 1: Our estimates are motivated by recent developments in non-Newtonian fluidmechanics and elliptic problems with non-standard growth conditions, and are the natural, ‘‘non-linear’’ counterpart of those obtained by Diening and Ru ˚z ˇic ˇka [12] in the linear case. 1. Introduction In recent years, increasing attention has been paid to the study of the so called gen- eralized Lebesgue spaces L pðxÞ ðW; R N Þ, that is L pðxÞ ðW; R N Þ :¼ n f : W ! R N : f is measurable and Ð W j f j pðxÞ dx < y o ð1Þ where W H R n is a bounded domain and p : W !ð1; þyÞ is in general taken to be a con- tinuous function (there is no obstruction in taking a more general pðxÞ, but the resulting space has very few properties if no geometric condition on p is imposed). The Luxemburg type norm k f k L pðxÞ ðW; R N Þ :¼ inf  l > 0 : Ð W f l         pðxÞ dx e 1  makes L pðxÞ a Banach space. Accordingly, the generalized W 1; pðxÞ ðW; R N Þ spaces are de- fined by W 1; pðxÞ ðW; R N Þ :¼fu A L pðxÞ ðW; R N Þ : Du A L pðxÞ ðW; R nN Þg;