## Abstract

We prove Calderón and Zygmund type estimates for a class of elliptic problems whose model is the non-homogeneous *p* (*x* )-Laplacean system

Under optimal continuity assumptions on the function *p* (*x* ) > 1 we prove that

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 Růžička [*L. Diening and M. Růžička*, Calderón-Zygmund operators on generalized Lebesgue spaces *L*
*
^{p}
*

^{(‧)}and problems related to ﬂuid dynamics, J. reine angew. Math. 563 (2003), 197–220] in the linear case.

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