Local spectral convergence inRCD∗(K,N)spaces

Abstract

In this note we give necessary and sufficient conditions for the validity of the local spectral convergence, in balls, on the RCD * -setting.

Collapsed Ricci Limit Spaces as Non-Collapsed RCD Spaces

“…5 Such embedding maps are introduced and studied first in [11] for closed Riemannian manifolds. Recently in [7] this observation is generalized to RCD(K, N ) spaces by using stability results of Sobolev functions with repect to the pmGH convergence proved in [5,6]. It seems to the author that this technique, using a geometric flow, is useful for all conjectures proposed in this paper even in the case when (X, d) is non-compact.…”

Weyl’s law on $RCD^{\ast} (K, N)$ metric measure spaces

“…Remark that an L 2 -convergence theorem for global heat flows on (X j , d j , µ j ) was proved in [23]. An H 1convergence theorem for local heat flows has been recently obtained by Ambrosio-Honda in [7]. Lemma 3.9.…”

“…(2) The space W 1,p 0 (Ω) is equivalent to the spaceĤ 1,p 0 (Ω) given in [7] by Ambrosio-Honda. Let (X, d, µ) be an RCD * (K, N) metric measure space with some K ∈ R and some N 1.…”

A Sufficient Condition to a Regular Set Being of Positive Measure on Spaces

“…Before stating the main result in this section, we give the following definitions and results developed by Ambrosio and Honda [4]. For an open set A ⊂ X, LIP c (A, d) denotes the set of all Lipschitz functions whose supports are compact and contained in A.…”

“…for any x ∈ X and for every R > 0 excepting at most countably many positive numbers (see [4,Lemma 2.12]). Analogously the local Cheeger energy is defined as…”