We make explicit the p-dependence of C in the gradient estimate \(\parallel \nabla u \parallel_{\infty}^{p-1}\ \leq\ C \parallel f\parallel_{N,1}\) by Cianchi and Maz’ya (2011). In such inequality, the constant C is uniform with respect to f ∈ LN,1(Ω), and u is the weak solution to the Poisson equation −div(∣∇u∣p−2∇u) = f in a bounded domain Ω ⊂ ℝN, N ≥ 3, coupled with either Neumann or Dirichlet homogeneous boundary conditions. The case N = 2 with f ∈ Lq(Ω), for some q > 2, is also considered.
