Fractional Laplacian, homogeneous Sobolev spaces and their realizations

Abstract

We study the fractional Laplacian and the homogeneous Sobolev spaces on R d , by considering two definitions that are both considered classical. We compare these different definitions, and show how they are related by providing an explicit correspondence between these two spaces, and show that they admit the same representation. Along the way we also prove some properties of the fractional Laplacian.

Fractional Paley–Wiener and Bernstein spaces

“…We remark that for f ∈ S the fractional Laplacian Δ s 2 f is a well-defined function and that ‖ ⋅ ‖ s,p is a norm on the Schwartz space (see, for instance, [18]). Therefore, we define the homogeneous Sobolev space Ẇ s,p as the closure of S with respect to ‖ ⋅ ‖ s,p , i.e., As described in [18], the space Ẇ s,p turns out to be a quotient space of tempered distributions modulo polynomials of degree m = ⌊s − 1∕p⌋ , where we denote by ⌊x⌋ the integer part of x ∈ ℝ and by P m the set polynomials of degree at most m, where m ∈ ℕ 0 . In [18,Corollary 3.3] we prove that f ∈Ẇ s,p if and only if…”

“…In order to avoid working in a quotient space, instead of considering the spaces Ẇ s,p , we consider the realization spaces E s,p , see [18,Corollary 3.2]. Inspired by the works of G. Bourdaud [10][11][12], if m ∈ ℕ 0 ∪ {∞} and Ẋ is a given subspace of S � ∕P m which is a Banach space, such that the natural inclusion of Ẋ into S � ∕P m is continuous, we call a subspace E of S ′ a realization of Ẋ if there exists a bijective linear map such that R[u] = [u] for every equivalence class [u] ∈Ẋ .…”

“…We recall the definition of the homogeneous Lipschitz space Λ , for > 0 , for details see e.g. [15] and [18]. For k ∈ ℕ and h ∈ ℝ ⧵ {0} , let D k h , the difference operator of increment h and of order k, be defined as follows.…”

“…Proof By the results in [18], since f 0 ∈ E s,2 , there exists a sequence { n } ⊆ S such that {Δ s 2 n } is a Cauchy sequence in L 2 , and n → f 0 in S � ∕P m , where m = ⌊s − 1 2 ⌋ , that is, ⟨ n , ⟩ → ⟨f 0 , ⟩ = ⟨f 0 ,̂ ⟩ , as n → ∞ , for all ∈ S m . Therefore, as n → ∞ , for all ∈Ŝ m = S ∩ { ∈ S ∶ P ;m;0 } = 0 .…”

“…As in the proofs of Theorems 1 and 2, we see that Φ n ∈ PW s a . Moreover, using [18,Corollary 3.4] and the fact that n ∈ S ∞ , we see that…”