# Harnack inequality for fractional laplacian

Let u be a positive solution of $s\in (0, 1)$ \left\{\begin{aligned} (-\Delta )^{s} u &= 0 \text{ in } (-2T, 2T)\\ u &=g\quad\text{in}\quad \mathbb R\setminus(-2T, 2T). \end{aligned} \right. If $u \in C^2(-T, T)\cap C^1 [-T, T]$ and $g$ is a bounded positive function on $\mathbb R$, is there a Harnack inequality in one dimension for the above kind of equation.

By the way, in your case \begin{aligned} u(x) & = \frac{1}{\Gamma(1 + s) \lvert\Gamma(-s)\rvert} \int_{\mathbb{R} \setminus (-2T,2T)} \biggl( \frac{4 T^2 - x^2}{y^2 - 4 T^2}\biggr)^{\!s} \frac{1}{|x - y|} \, g(y) dy \\ & \qquad + (4 T^2 - 1)^s \biggl(\frac{c_1}{2 T - x} + \frac{c_2}{2 T + x}\biggr) \end{aligned} for $x \in (-2T, 2T)$ for some constants $c_1, c_2 \geqslant 0$ by the result of K. Bogdan or Z.-Q. Chen and R. Song (see also [Hmissi, Fonctions harmoniques pour les potentiels de Riesz sur la boule unite, Exposition. Math. 12(3) (1994): 281–288]), without assuming any smoothness of $u$.