Sobolev Spaces on Locally Compact Abelian Groups : Compact Embeddings and Local Spaces

We continue our research on Sobolev spaces on locally compact abelian (LCA) groups motivated by our work on equations with infinitely many derivatives of interest for string theory and cosmology. In this paper, we focus on compact embedding results and we prove an analog for LCA groups of the classical Rellich lemma and of the Rellich-Kondrachov compactness theorem. Furthermore, we introduce Sobolev spaces on subsets of LCA groups and study its main properties, including the existence of compact embeddings into L-spaces.


Introduction
In this paper we continue our research on Sobolev spaces on locally compact abelian groups [1,2], and we examine analogs of the Rellich lemma and the Rellich-Kondrachov compactness theorem.Sobolev spaces are well understood on domains of R  ; see [3,4], compact Riemannian manifolds [5,6], and metric measure spaces [7][8][9].There are also some works on Sobolev spaces in the -adic context; see [10,11] and references therein and in special cases of locally compact groups such as the Heisenberg group [12].
We are interested in Sobolev spaces in this general context due to our work on nonlinear equations "in infinitely many derivatives" of interest for contemporary physical theories: in [13][14][15], two of the present authors in collaboration with H. Prado have investigated the existence of regular solutions to the generalized Euclidean Bosonic string equation Δ −Δ  =  (, ) ,  > 0 (1) and some of its generalizations, and, in [16,17], the same researchers have developed a functional calculus appropriate for the study of the initial value problem for "ordinary" equations of the form  (  )  =  () . ( Equations such as (1) and ( 2) are specially interesting for string theory and cosmology; see [18][19][20][21] and references therein.These two areas are undergoing such a fast development that it seems important to understand (1) and (2) in contexts beyond the usual geometric arena of analysis on (Riemannian) manifolds.We think that topological groups are a natural testing ground for gathering a better understanding of ( 1) and (2).For instance, this setting would allow us to consider (1) for functions  on finite spaces with group structure (see, e.g., [22]), or for functions depending on an infinite number of independent variables.On the other hand, this generalization makes it necessary to develop a theory of Sobolev spaces on LCA groups appropriate for the study of nonlocal equations along the lines of [13][14][15].It is indeed possible to do so, essentially because of the existence of group structure and the availability of Fourier transform.We introduced Sobolev spaces on LCA groups in [1].In that reference, we proved analogs of the Sobolev embedding and Rellich-Kondrachov theorems, and we used these results to prove the existence of regular solutions to (1) on compact abelian groups.Then in [2], we considered a version of the classical Rellich lemma and presented another theorem on regular solutions to (1).Now, our version of the Rellich lemma appearing in [2] relies on a technical assumption on the structure of the group of characters of the given group  which limits its applicability.In this paper, we remove this assumption and prove a version of the Rellich lemma which can be applied in great generality, and we also improve our original Rellich-Kondrachov theorem proven in [1].Moreover, we introduce Sobolev spaces on subsets of LCA groups, in analogy with the Sobolev spaces defined on domains of R  .As in this classical case, we expect these spaces to be useful in the study of differential equations and other applications [23].
We organize this paper as follows.In Section 2, we recall our definition of Sobolev spaces and our previous embedding and compactness results.In Section 3, we state and prove our new compactness results, and in Section 4, we discuss Sobolev spaces on subsets of LCA groups.
We use standard notations from harmonic analysis [24,25].Let us fix a locally compact abelian group .We denote by  the unique Haar measure of  and by  ∧ the dual group of the group ; that is,  ∧ is the locally compact abelian group of all continuous group homomorphisms from  to the circle group .The   spaces over  are defined as usual: and we set We also recall that the Fourier transform on  is defined as follows: if  ∈  1 (), then its Fourier transform is the function f :  ∧ → C given by We consider general LCA groups in Section 2, but we restrict ourselves to compact abelian groups when proving compactness results in Section 3.

Sobolev Spaces
We introduce Sobolev spaces following our previous papers [1,2].Our definition uses essentially the Fourier transform for LCA groups and, as explained in [1], it generalizes naturally the standard notions of Sobolev spaces in the case of T  and R  ; see [26] and [4,Chapter 4].
We denote by Γ the set Definition 1.Let us fix a map  ∈ Γ and a nonnegative real number .We will say that  ∈  2 () belongs to the Sobolev space    () if the following integral is finite: Moreover, for  ∈    (), its norm ‖‖    () is defined as follows: Remark 2. A particular instance of Definition 1 appears in the paper [26] by Feichtinger and Werther.Another particular case of our definition is in [27].We also note that in -adic analysis, Sobolev spaces are defined in a way analogous to our Definition 1: if we take () = ‖‖  , where ‖ ⋅ ‖  is a -adic norm on Q   ≃ Q ∧  , then ( 7) and ( 8) allow us to recover the -adic Sobolev spaces considered in [11].
Remark 3. The fact that  ∈ Γ implies that our spaces    () are Banach algebras under some assumptions on ; see our previous paper [1].
We recall the following results proven in [1].

Proposition 4.
Let  be a locally compact abelian group.One has the following.

Compact Embedding
We recall that the notation  →→  means that the space  is compactly embedded in .We begin with our refined version of the classical Rellich lemma.Theorem 6.Let  be a compact group.If lim  → ∞ () = ∞, that is, for each  > 0 there exists finite set  such that for any  ∈  ∧ \ , one has () ≥ , then for all  > , Proof.We begin the proof stating a classical fact (see, e.g., [28]) on the characterization of precompact sets in   () spaces.
Theorem 7 (see [28]).Let Φ be a family of functions in   (), 1 ≤  < ∞.Then Φ is compact in   () if and only if the following conditions hold.
(ii) For every  > 0, there exists compact set  in  such that for each  ∈ Φ, (iii) For all  > 0, there exists unit neighborhood  such that for all  ∈  and all  ∈ Φ,        −       () ≤ .
Now we return to the proof of Theorem 6.Let   be a bounded sequence in    (), ‖  ‖    () ≤ .We need to show that there exists subsequence that converges strongly in    ().We will prove this fact by showing that the following sequence is compact in  2 ( ∧ ).We use Weil's theorem: since  < , we get and so (i) holds.Now we consider condition (ii).Let us fix  > 0. We consider two cases: if  ∧ is finite group, then we can take simply  =  ∧ and condition (ii) is satisfied.On the other hand, if the dual group  ∧ is infinite, it is enough to recall that if  is a compact group, then by the Pontryagin duality theorem, its dual  ∧ is discrete and therefore every compact set must be finite.From our assumption, we can find a compact set  such that Hence, and so (ii) holds.It remains to check condition (iii).Since  ∧ is discrete and each set is open, we can take  = {}, where  is unit in  ∧ .Thus, condition (iii) is satisfied and Theorem 6 follows from Weil's result.
Theorem 6 appears in our previous paper [2] under the additional assumption that the dual group  ∧ is countable.Now we consider embeddings of    () into () and   ().We proved in [1] that    () is continuously embedded in ().We prove in Theorem 10 below that if  is compact, then    →→ (), and finally in Theorem 11, we consider a version of the Rellich-Kondrachov which refines an analogous result from [1].We need the following lemma.Lemma 8. Let  be a discrete group and  ∈  1 ().Then for every  > 0, there exists a finite set  such that for any ℎ ∈  \ , one has |(ℎ)| ≤ .
Remark 9. We note that if  is countable, the proof of Lemma 8 is elementary: The result then follows.

Sobolev Spaces on Subsets of LCA Groups
In this section, we deal with Sobolev spaces defined on subsets of locally compact abelian groups.As mentioned in Section 1, we are motivated by analogous studies of function spaces on domains of R  (see, e.g., [29]) by the fact that interesting applications exist, [23], and by the possibility of using them as tools for the study of differential equations on subsets of LCA groups.We start with the following definition.
Definition 12. Let  be a subset of a LCA group .We define the Sobolev space    () ⊂  2 () as the space of all  ∈  2 () such that there exists  ∈    () with |  =  and we equip it with the norm An analogous definition (of spaces    on domains of R  ) appears in [29]; see his Definition 2.3.It can be easily shown that    () is a Banach space.We will say that it is a local Sobolev space.
Using appropriate embeddings for    () and the definition of    (), we can prove the following.
Theorem 13.Let  be a locally compact abelian group and let  ⊂ .Then we have. ( We now prove the following compactness theorem in detail. Theorem 14.Let  be an LCA group and let  be a subset of  of finite measure.Assume that 1/(1 +  2 (⋅)) ∈   ( ∧ ) for some  >  and that      (ℎ) − 1 Then, for all  <  * , one has the compact embedding The convergence concept used in (34) is explained in our previous paper [1].Let us mention that a condition similar to (34) appears in the characterization of precompact sets in  2 () via Fourier transform; see [30].
Proof.We will need two lemmas which we proved in [1].
We conclude that   →  in  2 ().In order to finish the proof, it is enough to use the Vitali convergence theorem.We conclude that   →  in   () for  <  * .