Fractional Sobolev extension and imbedding

Abstract

Let Ω \Omega be a domain of R n \mathbb {R}^n with n ≥ 2 n\ge 2 and denote by W s , p ( Ω ) W^{s,\,p}(\Omega ) the fractional Sobolev space for s ∈ ( 0 , 1 ) s\in (0,\,1) and p ∈ ( 0 , ∞ ) p\in (0, \,\infty ) . We prove that the following are equivalent: (i) there exists a constant C 1 > 0 C_1>0 such that for all x ∈ Ω x\in \Omega and r ∈ ( 0 , 1 ] r\in (0,\,1] , | B ( x , r ) ∩ Ω | ≥ C 1 r n ; \begin{eqnarray*} |B(x,\,r)\cap \Omega |\ge C_1 r^n; \end{eqnarray*} (ii) Ω \Omega is a W s , p W^{s,\,p} -extension domain for all s ∈ ( 0 , 1 ) s\in (0,\,1) and all p ∈ ( 0 , ∞ ) p\in (0,\,\infty ) ; (iii) Ω \Omega is a W s , p W^{s,\,p} -extension domain for some s ∈ ( 0 , 1 ) s\in (0,\,1) and some p ∈ ( 0 , ∞ ) p\in (0,\,\infty ) ; (iv) Ω \Omega is a W s , p W^{s,\,p} -imbedding domain for all s ∈ ( 0 , 1 ) s\in (0,\,1) and all p ∈ ( 0 , ∞ ) p\in (0,\,\infty ) ; (v) Ω \Omega is a W s , p W^{s,\,p} -imbedding domain for some s ∈ ( 0 , 1 ) s\in (0,\,1) and some p ∈ ( 0 , ∞ ) p\in (0,\,\infty ) .