Improvement of Besov regularity for solutions of the fractional Laplacian

HUGO AIMAR; GASTÓN BELTRITTI; IVANA GOMEZ

CONSTRUCTIVE APPROXIMATION

SPRINGER

Lugar: Berlin; Año: 2015 vol. 41 p. 219 - 229

0176-4276

We prove a mean value formula for weak solutions of $hbox{div}(|y|^{a}hbox{grad}\u)=0$ in $mathbb{R}^{n+1}={ (x,y): xin mathbb{R}^{n}, yinmathbb{R} }$, $-1<a<1$ and balls centered at points of the form $(x,0)$. We obtain an explicit nonlocal kernel for the mean value formula for solutions of $(-riangle)^{s}f=0$ on a domain $D$ of $mathbb{R}^{n}$. When $D$ is Lipschitz we prove a Besov type regularity improvement for the solutions of  $(-riangle)^{s}f=0$.