On the Capacities Associated with Local Muckenhoupt Weights

We develop a theory of capacities associated with local Muckenhoupt weights. Fundamental properties of local Muckenhoupt weights will be revisited. Weak type boundedness of nonlinear potential and capacitary strong type inequalities associated with such weights will be addressed. The boundedness of the local maximal function on the spaces of Choquet integrals associated with such weighted capacities will be justified as an application of the theory. We also address on the thinness of sets with the Kellogg property as another application.