On the strong maximum principle for a fractional Laplacian

In this paper, we obtain a version of the strong maximum principle for the spectral Dirichlet Laplacian. Specifically, let $$d \in \{1,2,3,\ldots \}$$ , $$s \in (\frac{1}{2},1)$$ , and $$\Omega \subset \mathbb {R}^d$$ be open, bounded, connected with Lipschitz boundary. Suppose $$u \in L^1(\Omega )$$ satisfies $$u \ge 0$$ a.e. in $$\Omega $$ and $$(-\Delta )^s u$$ is a Radon measure on $$\Omega $$ . Then u has a quasi-continuous representative $${\tilde{u}}$$ . Let $$a \in L^1(\Omega )$$ be such that $$a \ge 0$$ a.e. in $$\Omega $$ . Then if $$\begin{aligned} (-\Delta )^s u + au \ge 0 \quad \text {a.e.} \text { in } \Omega \end{aligned}$$ and $${\tilde{u}} = 0$$ on a subset of positive $$H^s$$ -capacity of $$\Omega $$ , then $$u = 0$$ a.e. in $$\Omega $$ .