Suppose $(-\Delta)^s u=f \geq 0$ in a ball $B\_2$ and $u=0$ in $ R^N \setminus B\_2.$ Also suppose $u$ is $C^{s}$ non-negative and $(-\Delta)^s u=0$ in $B\_2 \setminus B\_1$ and $u\leq a$ on $\partial B\_1$ where $B\_1, B\_2$ is a ball of radius $1$ and $2$ and $a$ is a positive constant. Can one claim that $u\leq a$ in $B\_2 \setminus B\_1$ or can any estimate be obtained on the upper bound of $u$. In the case $s=1$, this is maximum principle.



fractional Laplacian estimates
**Yes**.


We have
$$ u(x) = \int\_{B\_1} G\_{B\_2}(x,y) f(y) dy , $$
where
$$ G\_{B\_2}(x,y) = C\_{N,s} \frac{1}{|x - y|^{N - 2s}} \int\_0^{T(x,y)} \frac{t^{s-1}}{(t+1)^{N/2}} dt , \\ T(x, y) = \frac{(4-|x|^2)(4-|y|^2)}{4|x-y|^2} $$
is the corresponding Green function. Fortunately, $1/|x - y|^{N-2s}$ and $T(x,y)$ are radially decreasing on $B\_2 \setminus B\_1$, so indeed the maximal value of $u$ over $B\_2 \setminus B\_1$ is taken somewhere over $\partial B\_1$. 


