Quasi-Continuous Vector Fields on RCD Spaces

Abstract

In the existing language for tensor calculus on RCD spaces, tensor fields are only defined m-a.e.. In this paper we introduce the concept of tensor field defined '2-capacity-a.e.' and discuss in which sense Sobolev vector fields have a 2-capacity-a.e. uniquely defined quasicontinuous representative.

Constancy of the dimension in codimension one and locality of the unit normal on $\mathrm{RCD}(K,N)$ spaces

“…The original approach to this matter (where normed modules were defined over a metric space endowed with a Borel measure) is not sufficient for our purposes, since we would like to work also with vector fields defined capacity-a.e. as in [DGP21].…”

“…Accordingly, we will propose in Definition 2.1 below a notion of normed module which unifies the two theories studied in [G18] and [DGP21].…”

“…We know that Cap satisfies (2.3) and m ≪ Cap, cf. [DGP21]. Given any µ as in (2.3), we define L 0 (µ) as the space of all the equivalence classes, up to µ-a.e.…”

“…It is not clear to us whether a dimensional decomposition can be built for all L 0 (µ)normed L 0 (µ)-modules, when µ is as in (2.3) but is not a Borel measure. One difficulty is, for instance, the fact that we would need to take µ-essential unions; it is not clear how to do it (in the case where µ is the 2-capacity, this is pointed out in [DGP21]). However, this is not really relevant for our purposes, since on L 0 (Cap)-normed L 0 (Cap)-modules we are just interested to upper local dimension bounds, thus we will not investigate further in this direction.…”

Vector calculus for tamed Dirichlet spaces

“…in proving the above mentioned semigroup domination results. On RCD(K, ∞) spaces, K ∈ R -on which (0.11) has been proven in [32] in order to find "quasi-continuous representatives" of vector fields -this has been observed in [16].…”