Trace and extension theorems for Sobolev-type functions in metric spaces

Abstract

A. Trace classes of Sobolev-type functions in metric spaces are subject of this paper. In particular, functions on domains whose boundary has an upper codimension-θ bound are considered. Based on a Poincaré inequality, existence of a Borel measurable trace is proven whenever the power of integrability of the "gradient" exceeds θ. The trace T is shown to be a compact operator mapping a Sobolev-type space on a domain into a Besov space on the boundary. Su cient conditions for T to be surjective are found and counterexamples showing that surjectivity may fail are also provided. The case when the exponent of integrability of the "gradient" is equal to θ, i.e., the codimension of the boundary, is also discussed. Under some additional assumptions, the trace lies in L θ on the boundary then. Essential sharpness of these extra assumptions is illustrated by an example.

Traces and extensions of certain weighted Sobolev spaces on $\mathbb{R}^n$ and Besov functions on Ahlfors regular compact subsets of $\mathbb{R}^n$

“…We will show that Ω will satisfy the hypotheses of our paper with K playing the role of E and the ball B replaced by Ω. However, in this case we obtain a sharper result by combining the results of [BS] with [Ma,Theorem 1.1] to obtain the full range 0 < θ ≤ 1 − α+n−Q p in Theorem 1.1. From [Ah, Theorem 1] we know that the snowflake curve is a quasicircle.…”

“…Hence if the von Koch snowflake domain satisfies a strong local β-shell condition, then (Ω, µ α ) is doubling and supports a 1-Poincaré inequality when α > −β, and in addition from Lemma 4.3 we know that ν = H Q | K is 2 + α − Q-codimension regular with respect to µ α , and so by [Ma,Theorem 1.1] the conclusions of Theorem 1.1 and Theorem 1.2 hold for the von Koch domain and its boundary.…”

“…In this note we do not assume that R n \ E is a domain (the example of E being a Sierpiński carpet does not have its complement in R 2 be a domain) nor do we restrict ourselves to the case of p = 2. Indeed, when n ≥ 3, in considering R n , the choice of p = n is related to the quasiconformal geometry of R n \ E. In many cases, including the three examples considered here as well as the setting considered in [Ma,DFM1,DFM2], the weight ω β (x) ≃ dist(x, E) β .…”

“…For such sets, the measure µ α is a weighted Lebesgue measure given in (1) below. The trace and extension theorems for the example of the von Koch snowflake curve follow from the work of [Ma] once a family of measures µ α , 0 ≥ α > −β 0 for a suitable β 0 are constructed and shown to satisfy the hypotheses of [Ma,Theorem 1.1]. We do this construction in Section 4.…”

Quantitative characterization of traces of Sobolev maps

“…, which is reminiscent of an Alhfors upper codimension-1 bound (see [16,17,22], where a doubling condition is made separately). The quantity |σ| Lip denotes the Lipschitz constant of the map σ…”