Maximum principle of fractional Laplacian Suppose $u$ is a sign changing classical solution of the fractional Laplacian $$ (-\Delta) ^{s}  u     = 0     \; \text{in } \Omega;  
   u=g    \text{ in } \mathbb R^N  -\Omega .$$
(a)Is it true that $\|u\|_{L^{\infty}(\mathbb R^N)}\leq \|g\|_{L^{\infty}(\mathbb R^N -\Omega)}$ when $s\in (0, 1).$
(b) Is this also true for the operator $(-\Delta) ^{s}+I$ where $I$ is the identity.
 A: If $u$ is a solution of the above problem, then $u$ is said to be harmonic in $\Omega$ with respect to $(-\Delta)^s$. If $u$ is continuous and bounded in $\Omega$, and $\Omega$ satisfies the exterior cone condition, then
$$ u(x) = \int_{\mathbb{R}^N \setminus \Omega} g(y) P_\Omega(x, dy) ,$$
where $P_\Omega(x, \cdot)$ is a probability measure on the complement of $\Omega$. This measure is called the harmonic measure for $(-\Delta)^s$, and it has a nice probabilistic interpretation: it is the distribution of the isotropic $2s$-stable Lévy process started at $x$ and stopped at the time of first exit from $\Omega$. As a consequence, $|u(x)| \leqslant \|g\|_\infty$, as desired.
You can find the above result in Bogdan's 1997 paper [1], see Lemma 17 therein. For further discussion and other references, you may like to take a look at my survey paper [2], in particular — Theorem 7.2 therein.
[1] K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Stud. Math., 123(1) (1997), 43–80, available at http://matwbn.icm.edu.pl/ksiazki/sm/sm123/sm12313.pdf.
[2] M. Kwaśnicki, Fractional Laplace Operator and its Properties, in: A. Kochubei, Y. Luchko, Handbook of Fractional Calculus with Applications. Volume 1: Basic Theory, De Gruyter Reference, De Gruyter, Berlin, 2019, DOI:10.1515/9783110571622-007.
