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.



Maximum principle of fractional Laplacian
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](https://doi.org/10.1515/9783110571622-007).


