I would like to know if there exist a Harnack type inequality for the non-local operator of the form $$(-\Delta)^s u= f \text{ in } B\subset \mathbb R^N$$ with $0<s<1$ and $N \geq 1.$ Here $B$ is a unit ball and $f\not \equiv 0$ is bounded in $\mathbb R^N$ and $u$ is non-negative and smooth function in $\mathbb R^N$. It is also known that $\int\_{\mathbb R^N} \frac{u}{1+|x|^{N+2s}}dx$ is finite. Does it imply that $$\sup\_{B\_{1/2}} u \leq C(\inf\_{B} u+ \|f\|\_{L^{\infty}(B)})$$ where $C>0$ is dependent on $N$ and $s$ only and $B\_{1/2}$ is a ball of radius $1/2$. Any reference is welcome. 



Harnack inequaliity for the fractional Laplacian
The classical Harnack's inequality $\sup\_{B\_{1/2}} u\_1 \le C \inf\_B u$ for non-negative solutions of $(-\Delta)^s u\_1 = 0$ in $B$ goes back to M. Riesz's 1938 seminal paper.


The bound $\sup\_B |u\_2| \le C \|f\|\_\infty$ for solutions of $(-\Delta)^s u\_2 = f$ in $B$ with $u\_2 = 0$ in $B^c$ follows from comparison principle and the fact that $(-\Delta)^s w = -1$ for $w(x) = C (1 - |x|^2)\_+^{s/2}$.


By combining the two (with $u = u\_1 + u\_2$), we find that
$$ \sup\_{B\_{1/2}} u \leqslant \sup\_{B\_{1/2}} u\_1 + \sup\_{B\_{1/2}} |u\_2| \leqslant C \inf\_B u\_1 + C \|f\|\_\infty$$
and
$$ \inf\_B u \geqslant \inf\_B u\_1 - \sup\_B |u\_2| \geqslant \inf\_B u\_1 - C \|f\|\_\infty .$$
These two inequalities imply the desired estimate.


