In this paper, we study the boundary Hölder regularity for solutions to the fractional Dirichlet problem in unbounded domains with boundary \(\begin{aligned} {\left\{ \begin{array}{ll} {(-\triangle )^s}u(x) = g(x),& \text {in } \Omega ,\\ u(x)=0, & \text {in } \Omega ^c.\\ \end{array}\right. } \end{aligned}\) Existing results rely on the global \(L^{\infty }\) norm of solutions to control their boundary \(C^s\) norm, which is insufficient for blow-up and rescaling analysis to obtain a priori estimates in unbounded domains. To overcome this limitation, we first derive a local version of boundary Hölder regularity for nonnegative solutions in which we replace the global \(L^{\infty }\) norm by only a local \(L^{\infty }\) norm. Then as an important application, we establish a priori estimates for nonnegative solutions to a family of nonlinear fractional equations on unbounded domains with boundaries.
