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 β-dimensional mean oscillation for β ∈ (0,n] to regularity theory of critical exponent elliptic equations. We first show that functions with gradient in weak-Lⁿ are in BMO^β for any β ∈ (0,n], improving the classical result ∇u ∈ Lⁿ implies u ∈ BMO. We apply this result to the Poisson equation − Δu =div F with zero boundary conditions in a bounded C¹ domain to show that u ∈ BMOβ when F is in weak-Ln. Next, we consider the n-Laplace equation:(Formula presented) , with F ∈ L¹(Ω) and show that the classical result u ∈ BMO can be improved to u ∈ BMO^β. Finally, we consider the n-Laplace equation in the case when F ∈ L¹, div F = 0 and prove that for smooth domains Ω we have the estimate (Formula presented), where the constant C is independent of F.