Robust estimates for elliptic nonlocal operators on doubling spaces

We study weak Harnack inequality and a priori H\"older regularity of harmonic functions for symmetric nonlocal Dirichlet forms on metric measure spaces with volume doubling condition. Our analysis relies on three main assumptions: the existence of a strongly local Dirichlet form with sub-Gaussian heat kernel estimates, a tail estimate of the jump measure outside balls and a local energy comparability condition. We establish the robustness of our results, ensuring that the constants in our estimates remain bounded, provided that the order of the scale function appearing in the tail estimate and local energy comparability condition, maintains a certain distance from zero. Additionally, we establish a sufficient condition for the local energy comparability condition.