Bourgain–Brezis–Mironescu Approach in Metric Spaces with Euclidean Tangents

Abstract

In the setting of metric measure spaces satisfying the doubling condition and the (1, p)-Poincaré inequality, we prove a metric analogue of the Bourgain–Brezis–Mironescu formula for functions in the Sobolev space $$W^{1,p}(X,d,\nu )$$ W 1 , p ( X , d , ν ) , under the assumption that for $$\nu $$ ν -a.e. point the tangent space in the Gromov–Hausdorff sense is Euclidean with fixed dimension N.

A characterization of BV and Sobolev functions via nonlocal functionals in metric spaces

“…A similar result was proved previously in Ahlfors-regular spaces in [29]. Górny [18], resp. Han-Pinamonti [21], studied the problem in certain PI spaces that "locally look like" Euclidean spaces, resp.…”

“…These results correspond to certain choices of the mollifiers ρ i satisfying (1.1). In the current paper, our main goal is to study this problem for more general mollifiers ρ i , of which the mollifiers considered in [15,18,29] are special cases. Moreover, we consider domains Ω = X.…”

“…After giving definitions in Section 2 and some preliminary results in Section 3, we prove the two directions of (1.4) in Sections 4 and 5. In Section 6 we give corollaries to our main Theorem 1.3, showing that the mollifiers considered in [15] and [18], as well as other choices, can be handled as special cases. In Section 7 we give a counterexample demonstrating that we do not generally have…”

Korevaar-Schoen-Sobolev spaces and critical exponents in metric measure spaces

“…if the inequality holds with C = 1. It was recently proved in [15] (see also [23]) that under the additional condition that the tangent space in the Gromov-Hausdorff sense is Euclidean with fixed dimension, the limit exists if p > 1 and f ∈ KS 1,p (X) and is given by lim…”

A new integral decoupling property of sub-Riemannian heat kernels and some notable consequences

“…To provide the reader with a perspective on our results we note that if, as we have done above, one looks at Theorem B as a corollary of Theorem A, then the spherical symmetry of the approximate identities ρ ε (|x − y|), and therefore of the Euclidean heat kernel in (1.8), seems to play a crucial role in the dimensionless characterisations (1.9) and (1.10). With this comment in mind, we mention there has been considerable effort in recent years in extending Theorem A to various non-Euclidean settings, see [3,37,15,19,34,11,29,12,2,31] for a list, far from being exhaustive, of some of the interesting papers in the subject. In these works the approach is similar to that in the Euclidean setting, and this is reflected in the fact that the relevant approximate identities ρ ε either depend on a distance d(x, y), or are asymptotically close in small scales to the well-understood symmetric scenario of R n .…”