All functions are locally $s$-harmonic up to a small error

Abstract

We show that we can approximate every function $f\in C^k(\overline{B_1})$ by an $s$-harmonic function in $B_1$ that vanishes outside a compact set.

That is, $s$-harmonic functions are dense in $C_{\mathrm{loc}}^k$.This result is clearly in contrast with the rigidity of harmonic functions in the classical case and can be viewed as a purely nonlocal feature.