On planar SobolevLpm-extension domains

Abstract

For each m ≥ 1 and p > 2 we characterize bounded simply connected Sobolev L m p -extension domains Ω ⊂ R 2 . Our criterion is expressed in terms of certain intrinsic subhyperbolic metrics in Ω. Its proof is based on a series of results related to the existence of special chains of squares joining given points x and y in Ω.An important geometrical ingredient for obtaining these results is a new "Square Separation Theorem". It states that under certain natural assumptions on the relative positions of a point x and a square S ⊂ Ω there exists a similar square Q ⊂ Ω which touches S and has the property that x and S belong to distinct connected components of Ω \ Q.

Sobolev regularity of the Beurling transform on planar domains

“…From [Jon81], we know that uniform domains (and in particular, Lipschitz domains) are Sobolev extension domains for any indices n ∈ N and 1 ≤ p ≤ ∞. One can find deeper results in that sense in [Shv10] and [KRZ15]. The reader can consider n ∈ N and 1 < p < ∞ to be two given numbers along the whole text.…”

Product of extension domains is still an extension domain

“…It is easy to give examples of domains that fail to be extension domains, for example, the slit disk Ω := D 2 (0, 1) \ {(x 1 , 0) : 0 ≤ x 1 < 1}. In general, the extension property for a fixed Ω may depend on the value of p, see [13], [16] and [12].…”

On Stein's extension operator preserving Sobolev–Morrey spaces

“…Moreover, we refer to for a description of extension domains for certain Sobolev–Morrey spaces in the case , , . Finally, we refer to and the references therein for recent advances in the theory of extension operators.…”