A new Cartan-type property and strict quasicoverings when p = 1 in metric spaces

Abstract

In a complete metric space that is equipped with a doubling measure and supports a Poincaré inequality, we prove a new Cartan-type property for the fine topology in the case p = 1. Then we use this property to prove the existence of 1-finely open strict subsets and strict quasicoverings of 1-finely open sets. As an application, we study fine Newton-Sobolev spaces in the case p = 1, that is, Newton-Sobolev spaces defined on 1-finely open sets. * 2010 Mathematics Subject Classification: 30L99, 31E05, 26B30.

Discrete convolutions of $$\mathrm {BV}$$ functions in quasiopen sets in metric spaces

“…In quasiopen sets, the role of Lipschitz cutoff functions needs to be taken by Sobolev functions (often called Newton-Sobolev functions in metric spaces). A theory of Newton-Sobolev cutoff functions in quasiopen sets when p = 1 was developed in [29], analogously to the case 1 < p < ∞ studied previously in [7]. In the current paper we apply this theory to construct partitions of unity in quasiopen sets, and then we develop an analog of the discrete convolution technique in such sets.…”

“…Much less is known (even in Euclidean spaces) in the case p = 1, but certain results of fine potential theory when p = 1 have been developed by the author in metric spaces in [28,29,30]. In quasiopen sets, the role of Lipschitz cutoff functions needs to be taken by Sobolev functions (often called Newton-Sobolev functions in metric spaces).…”

Capacities and 1-strict subsets in metric spaces

“…In [30,Theorem 4.3] it is shown that when A is a point in fine-int D, then A is a 1-strict subset of D. Now our goal will be to show that despite Example 4.4, there are many other 1-strict subsets A of D. Our first result in this direction is the following. Then there exists η ∈ N 1,1 0 (W ) with 0 ≤ η ≤ 1 on X, η = 1 in a neighborhood of H, and…”

“…In the case p = 1, 1-strict subsets were studied, analogously to [7], in [30]. However, these papers left largely open the question of how to detect which sets are strict subsets.…”

Generalized Lipschitz numbers, fine differentiability, and quasiconformal mappings

“…The next lemma is given in [36,Proposition 4.6]; there it is given for the relative capacity cap 1 , but for the capacity Cap 1 the proof is almost the same.…”