Sharp capacity estimates for annuli in weighted $$\mathbf {R}^n$$ R n and in metric spaces

Abstract

We obtain estimates for the nonlinear variational capacity of annuli in weighted R n 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 R n . Indeed, to illustrate the sharpness of our estimates, we give several examples of radially weighted R n , which are based on quasiconformality of radial stretchings in R n .

The annular decay property and capacity estimates for thin annuli

“…and summing over all j shows that w is an A 1 -weight. Finally, using Proposition 10.8 in [5] again together with (3.1), we obtain for p > 1 and…”

“…One can check that this is the extreme case showing that the measure µ has the global η-AD property (and that η is optimal). By Proposition 10.8 in Björn-Björn-Lehrbäck [5], for p > 1 and 1 2 ≤ r < 1,…”

“…In this paper, we continue the study of sharp estimates for the variational capacity cap p (B r , B R ), which we started in Björn-Björn-Lehrbäck [5]. Therein we concentrated on the case 0 < 2r ≤ R, while in the present work we are interested in the case where the annulus B R \ B r is thin, that is, 0 < 1 2 R ≤ r < R. Assume for a moment that the measure µ is doubling and that the space X supports a p-Poincaré inequality.…”

Sphericalization and p-harmonic functions on unbounded domains in Ahlfors regular spaces

“…We are now ready to refine Lemma 4.5. (b) We need to determine when C p ({∞}) = 0, for which we will use results from Björn-Björn-Lehrbäck [10]. As µ a is Ahlfors Q-regular, it is also reverse-doubling, i.e.…”

“…since 2p − Q > 0, by (5.1). In the notation of [10], this means that the dimension sets of ( X,d,μ) at ∞ satisfy…”