Hajłasz–Sobolev imbedding and extension

The author establishes some geometric criteria for a Hajłasz–Sobolev M ˙ ball s , p -extension (resp. M ˙ ball s , p -imbedding) domain of R n 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 F ˙ p , ∞ s ( R 2 ) | Ω = M ˙ ball s , p ( Ω ) for some/all s ∈ [ α , 1 ) and p = ( 2 − α ) / ( s − α ) , where F ˙ p , ∞ s ( R 2 ) | Ω denotes the restriction of the Triebel–Lizorkin space F ˙ p , ∞ s ( R 2 ) on Ω.