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

Hindawi Publishing Corporation Journal of Function Spaces Volume 2014, Article ID 404738, 6 pages http://dx.doi.org/10.1155/2014/404738 Research Article Sobolev Spaces on Locally Compact Abelian Groups: Compact Embeddings and Local Spaces PrzemysBaw Górka,1 Tomasz Kostrzewa,1 and Enrique G. Reyes2 1 2 Department of Mathematics and Information Sciences, Warsaw University of Technology, Ul. Koszykowa 75, 00-662 Warsaw, Poland Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Casilla 307 Correo 2, Santiago, Chile Correspondence should be addressed to Przemysław Górka; Received 23 May 2013; Accepted 20 December 2013; Published 9 February 2014 Academic Editor: Kehe Zhu Copyright © 2014 Przemysław Górka et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 𝐿𝑝 -spaces. 1. 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 R𝑛 ; see [3, 4], compact Riemannian manifolds [5, 6], and metric measure spaces [7–9]. There are also some works on Sobolev spaces in the 𝑝-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 [13–15], two of the present authors in collaboration with H. Prado have investigated the existence of regular solutions to the generalized Euclidean Bosonic string equation Δ𝑒−𝑐Δ 𝜙 = 𝑈 (𝑥, 𝜙) , 𝑐>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 𝑓 (𝜕𝑡 ) 𝜙 = 𝐽 (𝑡) . (2) Equations such as (1) and (2) are specially interesting for string theory and cosmology; see [18–21] 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 𝜙 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 [13–15]. 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 2 Journal of Function Spaces group 𝐺 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 R𝑛 . 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 𝐺. We denote by 𝑑𝑥 the unique Haar measure of 𝐺 and by 𝐺∧ the dual group of the group 𝐺; that is, 𝐺∧ is the locally compact abelian group of all continuous group homomorphisms from 𝐺 to the circle group 𝑇. The 𝐿𝑝 spaces over 𝐺 are defined as usual: 󵄨𝑝 󵄨 𝐿𝑝 (𝐺) = {𝑓 : 𝐺 󳨀→ C : ∫ 󵄨󵄨󵄨𝑓 (𝑥)󵄨󵄨󵄨 𝑑𝑥 < ∞} , 𝐺 (3) and we set 󵄨𝑝 󵄩󵄩 󵄩󵄩 󵄨 󵄩󵄩𝑓󵄩󵄩𝐿𝑝 (𝐺) = (∫ 󵄨󵄨󵄨𝑓 (𝑥)󵄨󵄨󵄨 𝑑𝑥) 𝐺 1/𝑝 . (4) We also recall that the Fourier transform on 𝐺 is defined as follows: if 𝑓 ∈ 𝐿1 (𝐺), then its Fourier transform is the function 𝑓̂ : 𝐺∧ → C given by 𝑓̂ (𝜉) = ∫ 𝜉(𝑥)𝑓 (𝑥) 𝑑𝑥. 𝐺 (5) We consider general LCA groups in Section 2, but we restrict ourselves to compact abelian groups when proving compactness results in Section 3. 2. 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 𝑛 and R𝑛 ; see [26] and [4, Chapter 4]. We denote by Γ the set Moreover, for 𝑓 ∈ 𝐻𝛾𝑠 (𝐺), its norm ‖𝑓‖𝐻𝛾𝑠 (𝐺) is defined as follows: 1/2 󵄨2 󵄩󵄩 󵄩󵄩 2 𝑠󵄨 󵄩󵄩𝑓󵄩󵄩𝐻𝛾𝑠 (𝐺) = (∫ (1 + 𝛾(𝜉) ) 󵄨󵄨󵄨󵄨𝑓̂ (𝜉)󵄨󵄨󵄨󵄨 𝑑𝜉) . 𝐺∧ 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 𝑝-adic analysis, Sobolev spaces are defined in a way analogous to our Definition 1: if we take 𝛾(𝜉) = ‖𝜉‖𝑝 , where ‖ ⋅ ‖𝑝 is a 𝑝-adic norm on Q𝑛𝑝 ≃ Q𝑛∧ 𝑝 , then (7) and (8) allow us to recover the 𝑝-adic Sobolev spaces considered in [11]. Remark 3. The fact that 𝛾 ∈ Γ implies that our spaces 𝐻𝛾𝑠 (𝐺) are Banach algebras under some assumptions on 𝑠; see our previous paper [1]. We recall the following results proven in [1]. Proposition 4. Let 𝐺 be a locally compact abelian group. One has the following. (1) Consider 𝐻𝛾𝑠 (𝐺) 󳨅→ 𝐿2 (𝐺). Moreover, for each 𝑓 ∈ 𝐻𝛾𝑠 (𝐺), the following inequality holds: 󵄩 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑓󵄩󵄩𝐿2 (...truncated)