In this paper, we establish refined regularity estimates for nonnegative solutions to the fractional Poisson equation \( (-\Delta )^s u(x) =f(x),\,\, x\in B_1(0). \) Specifically, we have derived Hölder, Schauder, and Ln-Lipschitz regularity estimates for any nonnegative solution u, provided that only the local \(L^\infty \) norm of u is bounded. These estimates stand in sharp contrast to the existing results where the global \(L^\infty \) norm of u is required. Our findings indicate that the local values of the solution u and f are sufficient to control the local values of higher order derivatives of u. Notably, this makes it possible to establish a priori estimates in unbounded domains by using blow-up and rescaling argument. As applications, we derive singularity and decay estimates for solutions to some super-linear nonlocal problems in unbounded domains, and in particular, we obtain a priori estimates for a family of fractional Lane–Emden type equations in \(\mathbb {R}^n.\) This is achieved by adopting a different method using auxiliary functions, which is applicable to both local and nonlocal problems.
