On compact embeddings into L and fractional spaces

The study of the fractional Laplacian operator ( − Δ ) s in R N with Dirichlet boundary conditions gained enormous momentum through its identification with a Neumann operator in R N × ( 0 , ∞ ) = R + N + 1 , a method mainly introduced by Caffarelli and Silvestre. Since then, several other operators have been studied using this method. In general, a crucial question is attached to this method: is the embedding (in the trace sense) on the ground space L q ( R N ) compact? This question is very important when dealing with problems of existence of solutions. This paper aims to answer this question for some operators. Passing to an abstract setting, let X , Y be Hilbert spaces and A : X → X ′ a continuous and symmetric operator. We suppose that X is dense in Y and that the embedding X ⊂ Y is compact. In this paper we show some consequences of this setting for the study of the fractional operator attached to A in the extension setting Ω × ( 0 , ∞ ) or R + N + 1 . Being more specific, we will give some examples where the embedding of the extension domain into L 2 ( Ω ) is compact, even in the case Ω = R N .