I would like to know if there exist a Harnack type inequality for the nonlocal 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 nonnegative 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.

$\begingroup$ You can have $u$ as large as you wish, with $f$ equal to zero. A constant function $u$ is an example. $\endgroup$ – Mateusz Kwaśnicki Aug 9 at 19:33

$\begingroup$ @ Mateusz Kwaśnicki Thanks for pointing out the mistake. I have edited it. $\endgroup$ – GabS Aug 10 at 6:27
The classical Harnack's inequality $\sup_{B_{1/2}} u_1 \le C \inf_B u$ for nonnegative 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.