Pego theorem on compact groups

The Pego theorem characterizes the precompact subsets of the square-integrable functions on $\mathbb{R}^n$ via the Fourier transform. We prove the analogue of the Pego theorem on compact groups (not necessarily abelian).


Introduction
Characterizing precompact subsets is one of the classical topics in function space theory.It is well known that the Arzelà-Ascoli theorem characterizes a precompact subset of the space of continuous functions over a compact Hausdorff space.The celebrated Riesz-Kolmogorov theorem provides a characterization of precompact subsets of L p ‫ޒ(‬ n ).We refer to [8] for a historical account of it.Weil [14, page 52] extended it to the Lebesgue spaces over locally compact groups.See [7] for its extension to the Banach function spaces over locally compact groups.
In 1985, Pego [13] used the Riesz-Kolmogorov theorem to find a characterization of precompact subsets of L 2 ‫ޒ(‬ n ) via certain decay of the Fourier transform.
Theorem 1.1.[13, Theorems 2 and 3] Let K be a bounded subset of L 2 ‫ޒ(‬ n ).Then, the following are equivalent: as ω → 0, both uniformly for f in K .
An application of this theorem to information theory has been provided in [13].
Pego-type theorems have also been studied via the short-time Fourier and wavelet transforms [2], the Laplace transform [11] and the Laguerre and Hankel transforms [10].The Pego theorem has been extended to the locally compact abelian groups with some technical assumptions [5].Using the Pontryagin duality and the Arzelà-Ascoli theorem, the authors in [6] showed that the technical assumptions are redundant.For the L 1 -space analogue of the Pego theorem over locally compact abelian groups, see [12].
In Section 2, we present preliminaries on compact groups.In Section 3, using Weil's compactness theorem, we extend Theorem 1.1 to (not necessarily abelian) compact groups; see Theorem 3.4.

Fourier analysis on compact groups
Let G be a compact Hausdorff group.Let m G denote the normalized positive Haar measure on G. Let L p (G) denote the p-th Lebesgue space w.r.t. the measure m G .The norm on the space L p (G) is denoted by ∥ • ∥ p .
We denote by G the space consisting of all irreducible unitary representations of G up to the unitary equivalence.The set G is known as the unitary dual of G and is equipped with the discrete topology.Note that the representation space H π of π ∈ G is a complex Hilbert space and finite-dimensional.Denote by d π the dimension of H π .
Let B(H π ) denote the space consisting of all bounded linear operators on H π endowed with the operator norm.
Let f ∈ L 1 (G).The Fourier transform of f is defined by For more information on compact groups, we refer to [4; 9].Throughout the paper, G will denote a (not necessarily abelian) compact Hausdorff group.The identity of G is denoted by e.We will denote by I d π the d π ×d π identity matrix.

Pego theorem on compact groups
We discuss the characterization of precompact subsets of square-integrable functions on G in terms of the Fourier transform.We need the following definitions.
Let K ⊂ L p (G). Define K := { f : f ∈ L p (G)}.K is said to be uniformly L p (G)-equicontinuous if for any given ϵ > 0 there exists an open neighborhood O of e such that Let us begin with some important lemmas.
Proof.Let (e U ) U ∈ be a Dirac net on G; see [1, page 24].By the Riemann-Lebesgue lemma [9,Theorem 28.40], e U ∈ c 0 -π∈ G B(H π ).Then, there exists a finite set A ⊂ G such that Let f ∈ K .We denote by e U f the pointwise product of e U and f .Now, Then, applying the Hausdorff-Young inequality [9, Theorem 31.22],we get Therefore, using the Minkowski integral inequality, we obtain By [1, Lemma 1.6.5, page 24], we get that there exists U ∈ such that Proof.Let ϵ > 0. Since K has uniform ℓ p -π∈ G B p (H π )-decay, there exists a finite set A ⊂ G such that , f ∈ K .
Let f ∈ K and y ∈ G.Then, applying [9,Corollary 31.25],we obtain where M is a positive number such that , π ∈ A and y ∈ O.

Hence,
The following corollary is a generalization of [13, Theorem 1] studied on ‫ޒ‬ n , and [5, Theorem 1] and [3, Lemma 2.5] studied on locally compact abelian groups.This is also an improvement of the corresponding result on compact abelian groups in the sense that we do not assume boundedness of the subset of L 2 (G).
Proof.This is a direct consequence of Lemmas 3.1 and 3.2.□ Now, we present our main result, that is, the Pego theorem over compact groups.It is a consequence of the Weil theorem and above corollary.Theorem 3.4.Let K be a bounded subset of L 2 (G).Then, the following are equivalent: Proof.For any given ϵ > 0 we have that sup (ii) Let A be a finite subset of G. Assume that K is a bounded subset of the linear span of the set consisting of matrix entries [4, page 139] of elements in A. Note that the matrix entries are bounded functions.For f ∈ K , using the Schur orthogonality relations [4, Theorem 5.8] we obtain that Thus, K has uniform ℓ 2π ∈ G B 2 (H π )-decay.Hence, by Theorem 3.4, K is precompact.In particular, the convex hull of the set consisting of matrix entries of elements in A is precompact.
Therefore, (i) and (ii) are equivalent by the Weil theorem [14, page 52] (or see[7,  Theorems 3.1 and 3.3]).Further, (ii) and (iii) are equivalent by Corollary 3.3.□ The following gives an example of a setK ⊂ L 2 (G) which is not precompact but K is uniformly L 2 (G)-equicontinuous and K has uniform ℓ 2 -π∈ G B 2 (H π )-decay.Example 3.5.Consider the set K = {nχ G : n ∈ ‫}ގ‬ ⊂ L 2 (G) as given in [7, Example 4.2].Since K consists of only constant functions, it is clear that K is uniformly L 2 (G)-equicontinuous.By Corollary 3.3, K has uniform ℓ 2 -π∈ G B 2 (H π )-decay.Since K is not bounded, K is not precompact.Now, with the help of our main result Theorem 3.4, we show that certain subsets of L 2 (G) are precompact.Example 3.6.(i) Let r ∈ ‫.ޒ‬ Consider the set K = r n χ G : n ∈ ‫ގ‬ ⊂ L 2 (G).Since r n : n ∈ ‫ގ‬ is bounded and K consists of only constant functions, it follows that K is bounded and uniformly L 2 (G)-equicontinuous.Therefore, by Theorem 3.4, K is precompact.