BMO and gradient estimates for solutions of critical elliptic equations

In this paper, we explore several applications of the recently introduced spaces of functions of bounded [Formula: see text]-dimensional mean oscillation for [Formula: see text] to regularity theory of critical exponent elliptic equations. We first show that functions with gradient in weak-[Formula: see text] are in [Formula: see text] for any [Formula: see text], improving the classical result [Formula: see text] implies [Formula: see text]. We apply this result to the Poisson equation [Formula: see text] with zero boundary conditions in a bounded [Formula: see text] domain to show that [Formula: see text] when [Formula: see text] is in weak-[Formula: see text]. Next, we consider the [Formula: see text]-Laplace equation: [Formula: see text] with [Formula: see text] and show that the classical result [Formula: see text] can be improved to [Formula: see text]. Finally, we consider the [Formula: see text]-Laplace equation in the case when [Formula: see text], [Formula: see text] and prove that for smooth domains [Formula: see text] we have the estimate [Formula: see text] where the constant [Formula: see text] is independent of [Formula: see text].