Sharp capacity estimates for annuli in weighted R^n and in metric spaces

We obtain estimates for the nonlinear variational capacity of annuli in weighted Rn and in metric spaces. We introduce four different (pointwise) exponent sets, show that they all play fundamental roles for capacity estimates, and also demonstrate that whether an end point of an exponent set is attained or not is important. As a consequence of our estimates we obtain, for instance, criteria for points to have zero (resp. positive) capacity. Our discussion holds in rather general metric spaces, including Carnot groups and many manifolds, but it is just as relevant on weighted Rn. Indeed, to illustrate the sharpness of our estimates, we give several examples of radially weighted Rn, which are based on quasiconformality of radial stretchings in Rn.