Towards the $C^{p^\prime}$-Regularity Conjecture in Higher Dimensions

A longstanding conjecture in elliptic regularity theory inquires whether a W^{1,p} function whose p-laplacian is bounded is locally of class C^{1,\frac{1}{p-1}}. While it is well known that such functions are of class C^{1,\alpha} for some unknown 0 < α < 1, establishing the sharp estimate turns out to be a rather delicate problem. Quite recently, the authors managed to establish the conjecture in the plane. In this article, we address the conjecture in higher dimensions and confirm its validity in a number of other meaningful cases