Improvement of Besov regularity for solutions of the fractional Laplacian

Abstract

We prove a mean value formula for weak solutions of div(|y| agradu) = 0 in Rn+1 = {(x, y) : x ∈ Rn, y ∈ R}, −1 < a < 1, and balls centered at points of the form (x, 0). We obtain an explicit nonlocal kernel for the mean value formula for solutions of (−)s f = 0 on a domain D of Rn. When D is Lipschitz, we prove a Besov type regularity improvement for the solutions of (−)s f = 0.