# Hopf type lemma for fractional Laplacian

Let $$0 and $$u\in C^s(\mathbb R^N).$$ Does the Hopf type of maximum principle hold for s-super-harmonic function $$(-\Delta)^su\geq 0$$ in a smooth bounded domain $$\Omega\subset \mathbb R^N.$$

• Yes. If $u$ is non-negative, $s$-super-harmonic in $\Omega$, and zero outside $\Omega$, then $u(x) \geqslant c_{s,\Omega} (\operatorname{dist}(x,\Omega^c))^s$. A Google query on "Hopf lemma fractional Laplacian" returns at least three recent papers with these words in the title or abstract, but in fact it all goes back to the 1997 paper The boundary Harnack principle for the fractional Laplacian by K. Bogdan – Mateusz Kwaśnicki Dec 2 at 21:58