# Harnack inequaliity for the fractional Laplacian

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 $$0 and $$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.

• You can have $u$ as large as you wish, with $f$ equal to zero. A constant function $u$ is an example. – Mateusz Kwaśnicki Aug 9 at 19:33
• @ Mateusz Kwaśnicki Thanks for pointing out the mistake. I have edited it. – GabS Aug 10 at 6:27

## 1 Answer

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.