A note on properties of the restriction operator on Sobolev spaces

1 Preliminaries

We study properties of Sobolev spaces on a general (non-Lipschitz) open set Ω⊂ℝn. In our companion paper [3] we studied two types of spaces: those consisting of distributions on ℝn (specifically, H~s⁢(Ω), H̊s⁢(Ω), HΩ¯s, defined below), and those consisting of distributions on Ω itself (specifically, Hs⁢(Ω) and H0s⁢(Ω), again defined below). In this note we study properties of the restriction operator as a mapping between the two types of spaces. The results presented here, while elementary, do not seem to be available in the literature, which generally focuses on the more standard Lipschitz case (cf., e.g., [9]). As in [3], our motivation is the study of integral equations on non-Lipschitz sets. (For a concrete example, see [3, §4], where we consider boundary integral equation reformulations of wave scattering problems involving fractal screens.)

We begin by defining the Sobolev spaces involved. Our presentation follows that of [3], which in turn is broadly based on [9]. Given n∈ℕ, let 𝒟⁢(ℝn) denote the space of compactly supported smooth test functions on ℝn, and for any open set Ω⊂ℝn, let 𝒟⁢(Ω):={u∈𝒟⁢(ℝn):supp⁡u⊂Ω}. For Ω⊂ℝn, let 𝒟*⁢(Ω) denote the space of distributions on Ω (anti-linear continuous functionals on 𝒟⁢(Ω))^[1]. Let 𝒮⁢(ℝn) denote the Schwartz space of rapidly decaying smooth test functions on ℝn, and 𝒮*⁢(ℝn) the dual space of tempered distributions (anti-linear continuous functionals on 𝒮⁢(ℝn)). For u∈𝒮⁢(ℝn), we define the Fourier transform u^⁢(𝝃):=1(2⁢π)n/2⁢∫ℝne-i⁢𝝃⋅𝐱⁢u⁢(𝐱)⁢d𝐱, 𝝃∈ℝn, extending this definition to 𝒮*⁢(ℝn) in the usual way. We define the Sobolev space Hs⁢(ℝn) by

which is a Hilbert space with the inner product

For any -∞<s<t<∞, Ht⁢(ℝn) is continuously embedded in Hs⁢(ℝn) with dense image, and we have ∥u∥Hs⁢(ℝn)<∥u∥Ht⁢(ℝn) for all 0≠u∈Ht⁢(ℝn). Recalling that ℱ⁢(∂𝜶⁡u/∂⁡𝐱𝜶)⁢(𝝃)=(i⁢𝝃)𝜶⁢u^⁢(𝝃) and |𝜶|:=∑j=1nαj for a multi-index 𝜶∈ℕ0n, by Plancherel’s theorem it holds that

Hence, functions with disjoint support are orthogonal in Hm⁢(ℝn) for m∈ℕ0. But we emphasise that this is not in general true in Hs⁢(ℝn) for s∈ℝ∖ℕ0.

For a closed F⊂ℝn, we define HFs:={u∈Hs⁢(ℝn):supp⁢(u)⊂F}. The question of whether a given set E⊂ℝn can support nontrivial elements of Hs⁢(ℝn) will be important in what follows. This question was investigated in detail in [5], where we introduced the concept of s-nullity. (This concept is referred to by some authors as (2,-s)-polarity, see, e.g., [8, §13.2].)

There are many different ways to define Sobolev spaces on an open subset Ω⊂ℝn. In [3] we studied the following three spaces, all of which are closed subspaces of Hs⁢(ℝn), hence Hilbert spaces with respect to the inner product inherited from Hs⁢(ℝn):

here m⁢(⋅) denotes the Lebesgue measure on ℝn. (We note that H̊s⁢(Ω) can also be identified with the set of functions defined on Ω which can be extended by zero to produce functions of the same Sobolev regularity on the whole of ℝn, see Remark 2.4.) These three spaces satisfy the inclusions

(with H̊s⁢(Ω) present only for s≥0). If Ω is sufficiently smooth (e.g., C0), then the three sets coincide, but in general all three can be different (this issue is studied in [3, §3.5]).

Another way to define Sobolev spaces on an open set Ω is by restriction from Hs⁢(ℝn). For s∈ℝ, let

where U|Ω denotes the restriction of the distribution U to Ω in the standard sense (cf. [9, p. 66]). The inner product on Hs⁢(Ω) can be written as (u,v)Hs⁢(Ω):=(Qs⁢U,Qs⁢V)Hs⁢(ℝn) for u,v∈Hs⁢(ℝn), where U,V∈Hs⁢(ℝn) are such that U|Ω=u, V|Ω=v and Qs is the orthogonal projection Qs:Hs⁢(ℝn)→(HΩcs)⊥, see [3, §3.1.4]. It follows that the restriction operator

We also introduce the closed subspace of Hs⁢(Ω) defined by

The question of when Hs⁢(Ω) and H0s⁢(Ω) are equal is studied in detail in [3, §3.6].

For any open set Ω⊂ℝn, closed set F⊂ℝn and s∈ℝ, the following dual space realisations hold, in the sense of unitary isomorphism (see [3, §3.2]):

The duality pairings corresponding to these realisations are defined in terms of the duality pairing

which extends the L2⁢(ℝn) scalar product.

2 Properties of the restriction operator |Ω:Hs(ℝn)→Hs(Ω)

In this section we examine the relationship between the spaces H~s⁢(Ω), H̊s⁢(Ω), HΩ¯s⊂Hs⁢(ℝn), whose elements are distributions on ℝn, and the spaces Hs⁢(Ω) and H0s⁢(Ω), whose elements are distributions on Ω. The two types of spaces are linked by the restriction operator |Ω:Hs(ℝn)→Hs(Ω), and in this section we investigate some of its properties. In particular, we ask: for a given value of s and an appropriate pair of subspaces X⊂Hs⁢(ℝn), Y⊂Hs⁢(Ω), when is |Ω:X→Y (i) injective? (ii) surjective? (iii) a unitary isomorphism?

We start by recalling that |Ω:X→Hs(Ω) is continuous with norm at most one for any subspace X⊂Hs⁢(ℝn) and that |Ω:(HΩcs)⊥→Hs(Ω) is a unitary isomorphism.

For Ω being Lipschitz with bounded ∂⁡Ω, we have the following result, which states that |Ω:H~s(Ω)→H0s(Ω) is an isomorphism for certain values of s. (As in [3, 9], we say that Ω is Lipschitz if its boundary can be locally represented as the graph, suitably rotated, of a Lipschitz function from ℝn-1 to ℝ, with Ω lying only on one side of ∂⁡Ω.) The result for s≥0 is classical (see, e.g., [9, Theorem 3.33]); the extension to -1/2<s<0 is proved below (it is an immediate consequence of Lemma 2.7  (i) and [3, Corollary 3.29 (ix)]). In interpreting this result one should recall that for Ω being Lipschitz, it holds that H0s⁢(Ω)=Hs⁢(Ω) if and only if s≤1/2 [3, Corollary 3.29 (ix)] and also that H~s⁢(Ω)=H̊s⁢(Ω)=HΩ¯s for all s∈ℝ (see [3, Lemma 3.15], which follows from [9, Theorems 3.29 and 3.21]), with H̊s⁢(Ω) present only for s≥0.

We would like to understand to what extent this result generalises to non-Lipschitz Ω, and also how |Ω acts on the spaces H̊s⁢(Ω) and HΩ¯s in the case where these are not equal to H~s⁢(Ω). Some partial results in this direction are provided by the following lemma.

Lemma 2.2  (ii) and (iii) only deal with the case s≥0. In the next lemma we relate properties of the restriction operator acting on H~s⁢(Ω) for s and -s. In particular, this lemma allows us to infer the statement of Lemma 2.1 for -1/2<s<0 from the classical statement for 0<s<1/2 (recalling that H0s⁢(Ω)=Hs⁢(Ω) for Ω being Lipschitz and s≤1/2). For clarity, in this lemma and its proof we denote the restriction operator acting on H~s⁢(Ω) as |Ωs:H~s(Ω)→Hs(Ω). The proof of the lemma makes use of the fact that we can characterise the Banach space adjoint of |Ωs in terms of |Ω-s, using the dual space realisations (1.3).

We have seen that the restriction operator |Ω:H~s(Ω)→H0s(Ω) is an isomorphism for Ω being Lipschitz and for s>-1/2, s∉{1/2,3/2,…}. The next proposition shows that this result also extends to the case where Ω is a finite union of disjoint Lipschitz open sets, even when the union is not itself Lipschitz. Note that we do not require the closures of the constituent open sets to be mutually disjoint. The result therefore applies, for example, to the prefractal sets generating the Sierpinski triangle [3, Figure 4 (a)], which are finite unions of equilateral triangles touching at vertices.

For s≥0, we can rephrase the results of this section as follows. For any open set Ω, the restriction operator |Ω:H~s(Ω)→H0s(Ω) is continuous with norm one, is injective, has dense image, and the zero extension u↦uze is its right inverse on its image, i.e., uze|Ω=u for all u∈H~s⁢(Ω)|Ω. Furthermore, for s≥0, the following conditions are equivalent:

1. |Ω:H~s(Ω)→H0s(Ω) is an isomorphism,

2. |Ω:H~s(Ω)→H0s(Ω) is surjective,

3. the zero extension u↦uze is continuous H0s⁢(Ω)→H~s⁢(Ω),

4. there exists c>0 such that ∥Φ∥Hs⁢(ℝn)≥c⁢∥ϕ∥Hs⁢(ℝn) for all ϕ∈𝒟⁢(Ω) and Φ∈Hs⁢(ℝn) such that Φ|Ω=ϕ.

By Proposition 2.8, we know all these conditions hold for disjoint unions of Lipschitz open sets with bounded boundary and s∉{1/2,3/2,…}. But results about the surjectivity (or otherwise) of |Ω:H~s(Ω)→H0s(Ω) on more general Ω do not seem to be available in the literature and in this case we only know (by Lemma 2.2  iii) that the conditions above are true for s∈ℕ0. The following therefore appear to be open questions: For which Ω are (i)–(iv) true for all s>-1/2, s∉{1/2,3/2,…}? For which values of s are they satisfied for a given Ω?

3 The space (HΩcs)⊥ and minimal norm extensions

From (1.2), the elements of (HΩcs)⊥ are the extensions of the elements of Hs⁢(Ω) with minimal Hs⁢(ℝn) norm. In this short section we make some remarks on the nature of the elements of (HΩcs)⊥, and on minimal norm extensions. We also refer the reader to the related discussion in Remark 2.14 above.

For m∈ℕ0, the fact that functions with disjoint support are orthogonal in Hm⁢(ℝn) (cf. (1.1) and the sentence following it) implies that H̊m⁢(Ω)⊂(HΩcm)⊥. Thus, we have

which, by duality (1.3), implies that

In particular, minimal extensions from H-m⁢(Ω) to H-m⁢(ℝn) are supported in Ω¯. Hence, if u∈H-m⁢(Ω), then there exists U∈HΩ¯-m with U|Ω=u. Furthermore, given any such U, the minimal extension of u is given by Q-m⁢U=U+w, where w∈H∂⁡Ω-m.