An Orlicz-Besov Poincaré Inequality via John Domains

Denote byB˙⁎α,ϕ(Ω)the intrinsic Orlicz-Besov space, whereα∈R,ϕis a Young function, andΩ⊂Rnis a domain. Forα∈(-n,0)and optimalϕ, via John domains, we establish criteria for bounded domainsΩ⊂Rnsupporting an Orlicz-Besov Poincaré inequality.‖u-uΩ‖Ln/|α|(Ω)≤C‖u‖B˙⁎α,ϕ(Ω) ∀u∈B˙⁎α,ϕ(Ω).This extends the known criteria for bounded domains supporting Sobolev-Poincaré inequality and its fractional analogue.

Dichotomy of global capacity density in metric measure spaces

The variational capacity {\operatorname{cap}_{p}} in Euclidean spaces is known to enjoy the density dichotomy at large scales, namely that for every {E\subset{\mathbb{R}}^{n}},\inf_{x\in{\mathbb{R}}^{n}}\frac{\operatorname{cap}_{p}(E\cap B(x,r),B(x,2r))}% {\operatorname{cap}_{p}(B(x,r),B(x,2r))}is either zero or tends to 1 as {r\to\infty}. We prove that this property still holds in unbounded complete geodesic metric spaces equipped with a doubling measure supporting a p-Poincaré inequality, but that it can fail in nongeodesic metric spaces and also for the Sobolev capacity in {{\mathbb{R}}^{n}}. It turns out that the shape of balls impacts the validity of the density dichotomy. Even in more general metric spaces, we construct families of sets, such as John domains, for which the density dichotomy holds. Our arguments include an exact formula for the variational capacity of superlevel sets for capacitary potentials and a quantitative approximation from inside of the variational capacity.

On a decomposition of regular domains into John domains with uniform constants

We derive a decomposition result for regular, two-dimensional domains into John domains with uniform constants. We prove that for every simply connected domain Ω ⊂ ℝ2 with C1-boundary there is a corresponding partition Ω = Ω1 ⋃ … ⋃ ΩN with Σj=1NH1(∂Ωj\∂Ω)≤θ such that each component is a John domain with a John constant only depending on θ. The result implies that many inequalities in Sobolev spaces such as Poincaré’s or Korn’s inequality hold on the partition of Ω for uniform constants, which are independent of Ω.