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Compactness in Wiener Amalgams on Locally Compact Groups

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Compactness in Wiener Amalgams on Locally Compact Groups IJMMS 2003:55, 3503–3517 PII. S0161171203212205 http://ijmms.hindawi.com © Hindawi Publishing Corp. COMPACTNESS IN WIENER AMALGAMS ON LOCALLY COMPACT GROUPS S. S. PANDEY Received 18 December 2002 We study the compactness of bounded subsets in a Wiener amalgam whose local and global components are solid Banach function (BF) spaces on a locally compact group. Our main theorem provides a generalization of the corresponding results of Feichtinger. This paper paves the way for the study of compact multiplier op- erators on general Wiener amalgams on the lines of Feichtinger. 2000 Mathematics Subject Classification: 43A65, 46F05. 1. Introduction. The concept of Wiener amalgams on a locally compact group was introduced by Feichtinger [4] in 1977. In a series of brilliant papers [4, 5, 7, 8], Feichtinger has explored the vital role of Wiener amalgams in general harmonic analysis and its various applications. These spaces, in fact, describe the global and local behaviours of functions or distributions independently and provide very convenient generalizations of the classical function and se- quence spaces. Initially, in 1926, Wiener [21] studied a special case of amal- gam spaces on the real line in the form of mixed normed spaces. Feichtinger, therefore, in the above-mentioned papers, named them “Wiener-type spaces.” But in subsequent papers (cf. [11, 12]) he has used the nomenclature “Wiener Amalgams” and made a distinction from ordinary amalgam spaces (cf. [10, page 395]). Fournier and Stewart [16], in a survey article, have given in detail the construction of ordinary amalgams using Lp-spaces as local and Lq-spaces as global components and pointed out a number of applications in classical analysis. In order to define Wiener amalgams W (B,Y )(Rm) over m-dimensional Eu- clidean spaces, Feichtinger has used the spaces S(Rm) of tempered distribu- m tions as a reservoir to define the Bloc(R ) space. Since the notion of deriva- tion is not defined on locally compact groups, we cannot use the direct gen- eralization of the space of tempered distributions S(Rm). On the other hand the concept of Schwartz-Bruhat spaces (cf. [1]) of rapidly decreasing smooth functions imposes too stringent conditions on test functions, while their topo- logical dual spaces of tempered distributions, involving Frechet-type spaces, are too general for use in the real situations. Therefore, we follow the well- p known Feichtinger algebra track (cf. [6]) to develop a Banach space Ᏼw (Ᏻ) 3504 S. S. PANDEY ∼ p of test functions and the space of its antifunctionals Ᏼw (Ᏻ),withᏳ being a locally compact group and 1 ≤ p<∞, on the lines of Feichtinger and Gröchenig [15]. Moreover, by virtue of the inclusion relations ∼ Ᏼp ⊆ Ᏼ1 ⊆ Ᏼ1 ¬ ⊆ Ᏼp Ᏻ w w w w ( ), (1.1) ∼ p our reservoir spaces Ᏼw (Ᏻ) are larger than that of Feichtinger and Gröchenig [15]. We define the Wiener amalgam W(B,Y)(Ᏻ) on the lines of Feichtinger ∼ p m using the space Ᏼw (Ᏻ) in place of S(R ). The problem of compactness in Lp-spaces on locally compact groups was studied by Weil [20]. The corresponding results for Orlicz spaces have been discussed by Bund [2]. More general results on compactness in Köthe spaces have been obtained by Goes and Welland [17]. The compactness problems for various spaces of differentiable functions on the Euclidean spaces have been studied by Nikol’ski˘ı[18]. Feichtinger [9] has studied in detail the problem of compactness in trans- lation-invariant Banach spaces of distributions on locally compact groups. His theorems are more general than earlier results in this line of work and hold true even for the space of ultra distributions on locally compact groups. He has pointed out that compactness criteria hold for a family of much more general Banach spaces of distributions defined by decomposition methods, including the usual Besov spaces (cf. [13]). Feichtinger has also discussed in [13] the applications of his results for the study of compact multipliers on translation-invariant Banach spaces of functions or distributions on locally compact groups. The purpose of this paper is to study the compactness of bounded subsets of a general Wiener amalgam W(B,Y)(Ᏻ), where the local and global compo- nents B and Y are both solid Banach function (BF) spaces and satisfy some other suitable conditions. In Section 2, we give the necessary definitions and concepts for use in Section 3 which deals with the weighted Banach spaces, and in Section 4, we describe the partition of unity on Ᏻ.InSection 5, we define the Wiener amalgams W(B,Y)(Ᏻ) and in Proposition 5.2, we mention some useful properties of these spaces. In Section 6, we define spline quasi-interpolation operators and state two useful lemmas. Section 7 is devoted to the study of tightness and equicontinuity of a bounded subset in a Wiener amalgam space. Theorem 8.1 is the main result of this paper, which provide a generalization of the corresponding results of Feichtinger (cf. [9, Theorem 2.2] and [12, Propo- sition 5, page 131]). 2. Preliminaries. We suppose that Ᏻ is a locally compact group and Γ its dual group consisting of all continuous characters on Ᏻ.Wedenotebydx and dγ the normalized Haar measures on Ᏻ and Γ , respectively. We also assume that Ᏻ is a σ -compact group so that all sums, coverings, and index sets on Ᏻ COMPACTNESS IN WIENER AMALGAMS ON LOCALLY ... 3505 are countable. We define the left- and right- translation operators on Ᏻ by Lxf(y)= f(y−x) ∀x,y ∈ Ᏻ, −1 −1 (2.1) Rxf(y)= f xy ∆ (y), and the involution functions f˘ and f ∇ by f(x)˘ = f(−x), f ∇(x) = f(−x), (2.2) where ∇ denotes the Haar modular function on Ᏻ. We denote the Fourier transform of f on Ᏻ by fˆ such that f(γ)ˆ = (−x,γ)f(x)dx. (2.3) Ᏻ We denote by C0(Ᏻ) the space of all continuous functions on Ᏻ vanishing at infinity and by Cc (Ᏻ) the topological vector space of all continuous func- tions on Ᏻ with compact support. The space Cc (Ᏻ) is endowed with its natural inductive limit topology and its topological dual can be identified with R(Ᏻ), ∞ the space of all Radon measures on Ᏻ. Also, the closure of Cc (Ᏻ) in L (Ᏻ) is 0 Ᏻ 1 Ᏻ Ᏻ identified with C ( ). The space Lloc( ) of all locally integrable functions on is considered as a closed subspace of R(Ᏻ), that is, two measurable functions which coincide locally almost everywhere (l.a.e) are identical as usual. This im- 1 Ᏻ plies that the topology on Lloc( ) is generated by the system of seminorms {sK } given by sK (f ) = f(x) dx, (2.4) K where K varies over the family of all compact subsets of Ᏻ. 3. Weighted Banach spaces on Ᏻ. We assume that m : Ᏻ → R+ is a submul- tiplicative weight function on Ᏻ. It is well known that every submultiplicative weight function is locally bounded. A locally integrable function w : Ᏻ → R+ is called a right moderate weight function provided that there exists a submultiplicative weight function m on Ᏻ such that w(x+y) ≤ w(x)m(y) ∀x,y ∈ Ᏻ. (3.1) We also assume that all weight functions are symmetric and satisfy the non- quasianalyticity Beurling-Domer condition (BD condition) (cf. [19, Chapter VI, Section 3]). It can easily be seen that every moderate weight function is locally bounded. 3506 S. S. PANDEY p We denote by Lw (Ᏻ),1≤ p<∞, the Banach space of functions on Ᏻ with respect to the norm 1/p p p p f p,w ≡ f | Lw = f(x) w (x)dx < ∞. (3.2) Ᏻ =∞ ∞ Ᏻ In case p ,wedenotebyLw ( ) the Banach space of all measurable functions f on Ᏻ under the norm | ∞ = ∈ Ᏻ ∞ f Lw esssup f(x) w(x) : x < . (3.3) p Ᏻ p Ᏻ = p/(1−p) The conjugate space of Lw ( ) is the space Lw ( ), where w w and p 1/p+1/p = 1. It can easily be seen that Lw (Ᏻ),1<p<∞, is a reflexive Banach 1 Ᏻ space and Lw ( ) is a Banach algebra under convolution, which is known as Beurling algebra, p ∗ 1 ⊆ p Lw Lw Lw , (3.4) ∗ | p ≤ | 1 | p (g f) Lw g Lw f Lw ∈ 1 Ᏻ ∈ p Ᏻ for all g Lw ( ) and f Lw ( ). p Also, it can be verified that the space Lw (Ᏻ) is invariant under left- and right- translation operators Lx and Rx, respectively, and the operator norms satisfy the condition p p m(x) ≥ Lx | Lw , Rx | Lw . (3.5) Let (B, · B ) be a Banach space of measurable function on Ᏻ. The space B 1 Ᏻ is called a BF space provided that it is continuously embedded into Lloc( ). The space B is known as a solid BF space if it satisfies the following solidity ∈ ∈ 1 Ᏻ | |≤| | ⇒ ∈ ≤ condition: f B, g Lloc( ) with g(x) f(x) l.a.e. g B, g B f B . The Lp-spaces are simple examples of solid BF spaces. We write u(x) = Lx | B , = ∆ −1 | v(x) x Rx−1 B , (3.6) w(x) ≥ c max u(x),u x−1 ,v(x),v x−1 ∆ x−1 , 1 ≤ w(x) < ∞, w(x) = w x−1 ∆ x−1 . By virtue of the closed graph theorem, the operators Lx and Rx are bounded on B for each x in Ᏻ. Thus the maps w : x → |Lx | B| and w : x → |Rx | B| are well defined and submultiplicative on Ᏻ. 4. Partition of unity on Ᏻ.
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