LAUSR

In this paper, we study quasilinear elliptic equations with the nonlinearity modelled after the p(x) p ( x ) -Laplacian on nonsmooth domains and obtain sharp Calderon–Zygmund type estimates in the variable exponent setting. In a recent work of [12] , the estimates obtained were strictly above the natural exponent and hence there was a gap between the natural energy estimates and estimates above p(x) p ( x ) , see (1.3) and (1.4) . Here, we bridge this gap to obtain the end point case of the estimates obtained in [12] , see (1.5) . In order to do this, we have to obtain significantly improved a priori estimates below p(x) p ( x ) , which is the main contribution of this paper. We also improve upon the previous results by obtaining the estimates for a larger class of domains than what was considered in the literature.