Non-local Torsion functions and Embeddings

Original · EN

Given s ∈ (0,1), we discuss the embedding of Dˢ,ᵖ₀(Ω) in Lq(Ω). In particular, for 1≤ q < p we deduce its compactness on all open sets Ω⊂ Rⁿ on which it is continuous. We then relate, for all q up the fractional Sobolev conjugate exponent, the continuity of the embedding to the summability of the function solving the fractional torsion problem in Ω in a suitable weak sense, for every open set Ω. The proofs make use of a non-local Hardy-type inequality in Dˢ,ᵖ₀(Ω), involving the fractional torsion function as a weight.