Hajłasz–Sobolev imbedding and extension

The author establishes some geometric criteria for a Hajłasz–Sobolev -extension (resp. -imbedding) domain of Rn with n⩾2, s∈(0,1] and p∈[n/s,∞] (resp. p∈(n/s,∞]). In particular, the author proves that a bounded finitely connected planar domain Ω is a weak α-cigar domain with α∈(0,1) if and only if for some/all s∈[α,1) and p=(2−α)/(s−α), where denotes the restriction of the Triebel–Lizorkin space on Ω.