Sobolev Classes and Horizontal Energy Minimizers Between Carnot–carathéodory Spaces

2023

Preprint

We consider weakly and strongly asymptotically mean value harmonic (amv-harmonic) functions on subriemannian and RCD settings. We demonstrate that, in non-collapsed RCD-spaces with vanishing metric measure boundary, Cheeger harmonic functions are weakly amv-harmonic and that, in Carnot groups, weak amv-harmonicity equivalently characterizes harmonicity in the sense of the sub-Laplacian.In homogeneous Carnot groups of step 2, we prove a Blaschke-Privaloff-Zaremba type theorem. Similar results are discussed in the settings of Riemannian manifolds and for Alexandrov surfaces.

Section: Theorem 13mentioning

confidence: 97%

Asymptotically mean value harmonic functions in sub-Riemannian and RCD settings

Adamowicz¹,

Kijowski²,

2023

Preprint

“…Both RCD-spaces and Carnot groups have the property that the Korevaar-Schoen energy is a Dirichlet form. For RCD-spaces, this follows from [28] (in the noncollapsed case also from work of Sturm [52,53]), while for Carnot groups it is implied by [54]. See Propositions 3.6 and 3.3 for precise statements.…”

Section: Overview and Strategymentioning

confidence: 97%

Asymptotically Mean Value Harmonic Functions in Subriemannian and RCD Settings

Adamowicz

Kijowski²,

2023

J Geom Anal

We consider weakly and strongly asymptotically mean value harmonic (amv-harmonic) functions on subriemannian and RCD settings. We demonstrate that, in non-collapsed RCD-spaces with vanishing metric measure boundary, Cheeger harmonic functions are weakly amv-harmonic and that, in Carnot groups, weak amv-harmonicity equivalently characterizes harmonicity in the sense of the sub-Laplacian. In homogeneous Carnot groups of step 2, we prove a Blaschke–Privaloff–Zaremba type theorem. Similar results are discussed in the settings of Riemannian manifolds and for Alexandrov surfaces.

“…Both RCD-spaces and Carnot groups have the property that the Korevaar-Schoen energy is a Dirichlet form. For RCD-spaces, this follows from [31] (in the non-collapsed case also from work of Sturm [63,64]), while for Carnot groups it is implied by [65]. See Propositions 6.4 and 6.3 for precise statements.…”

Section: Introductionmentioning

confidence: 97%

Asymptotically mean value harmonic functions in doubling metric measure spaces

Adamowicz¹,

Kijowski²,

2020

Preprint

We consider weakly and strongly asymptotically mean value harmonic (amv-harmonic) functions on metric measure spaces which, in the classical setting, are known to coincide with harmonic functions.We demonstrate that, in non-collapsed RCD-spaces with vanishing metric measure boundary, Cheeger harmonic functions are weakly amv-harmonic and that, in Carnot groups, weak amv-harmonicity equivalently characterizes harmonicity in the sense of the sub-Laplacian. Moreover, in the first Heisenberg group, we prove a Blaschke-Privaloff-Zaremba type theorem which yields the equivalence of both weak and strong amv-harmonicity with harmonicity in the sense of the sub-Laplacian.In doubling metric measure spaces we show that strongly amv-harmonic functions are Hölder continuous for any exponent below one. More generally, we define the class of functions with finite amv-norm and show that functions in this class belong to a fractional Hajłasz-Sobolev space and their blow-ups satisfy the mean-value property.In the toy model case of weighted Euclidean domains, we identify the elliptic PDE characterizing amv-harmonic functions, and use this to point out that Cheeger harmonic functions in RCD-spaces need not be weakly amv-harmonic without the non-collapsing condition.