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Natural proof of the characterization of relatively compact families in Lp−spaces on locally compact groups Mateusz Krukowski Institute of Mathematics, Łódź University of Technology, Wólczańska 215, 90-924 Łódź, Poland e-mail: [email protected] July 13, 2020 Abstract In the paper we look for an elegant proof of the characterization of relatively compact families in Lp−spaces. At first glance, the suggested approach may seem convoluted and lengthy, but we spare no effort to argue that our proof is in fact more natural than the ones existent in the litera- ture. The key idea is that the three properties which characterize relative compactness in Lp−spaces (Lp−boundedness, Lp−equicontinuity and Lp−equivanishing) are “preserved” (or rather “inherited”) when the family F ⊂ Lp is convolved with a continuous and compactly supported function. This new family turns out to be relatively compact in C0−space and it remains to be demonstrated that relative compactness in C0−space implies relative compactness of the original family F. Keywords : Arzelà-Ascoli theorem, Kolmogorov-Riesz theorem, Weil theorem, Sudakov theorem, Young’s convolution inequality AMS Mathematics Subject Classification (2010) : 43A15, 46B50, 46E15, 46E30 1 Introduction Let us begin with a brief historical overview1: Andrey Kolmogorov (1903 − 1987) was arguably the arXiv:1801.01898v3 [math.FA] 10 Jul 2020 first person, who succeeded in characterizing relatively compact families in Lp(RN ), when 1 <p< ∞ and all functions are supported in a common bounded set2. A year later (in 1932) Jacob David Tamarkin (1888 − 1945) got rid of the second restriction3 and in 1933 Marcel Riesz (1886 − 1969), a younger brother of Frigyes Riesz (1880 − 1956), proved the general case4: Theorem 1. Let 1 6 p< ∞. A family F ⊂ Lp RN is relatively compact if and only if 1See [20] for a much more detailed account. 2See [26] for the original paper by Kolmogorov. 3See [38] for the original paper by Tamarkin. 4See [32] for the original paper by Riesz. 1 • F is Lp−bounded, i.e. there exists M > 0 such that ∀f∈F kfkp 6 M, • F is Lp−equicontinuous, i.e. for every ε> 0 there exists δ > 0 such that p ∀|y|<δ |f(x + y) − f(x)| dx 6 ε, RN f∈F Z • F is Lp−equivanishing, i.e. for every ε> 0 there exists R> 0 such that p ∀f∈F |f(x)| dx 6 ε. Z|x|>R In 1940, André Weil (1906 − 1998) published a book “L’intégration dans les groupes topologique”, in which he demonstrated that Theorem 1 holds true even if we replace RN with a locally compact Hausdorff group G5: Theorem 2. Let G be a locally compact Hausdorff group. A family F ⊂ Lp(G) is relatively compact if and only if • F is Lp−bounded, p • F is L −equicontinuous, i.e. for every ε > 0 there exists an open neighbourhood Ue of the neutral element e such that p ∀y∈Ue |f(xy) − f(x)| dx 6 ε, f∈F ZG • F is Lp−equivanishing, i.e. for every ε > 0 there exists K ⋐ G (which means that K is a compact subset of G) such that p ∀f∈F |f(x)| dx 6 ε. ZG\K Weil’s book is oftentimes cited as a reference source when it comes to characterizing relatively compact families in Lp−spaces6. However, the argument of the french mathematician is (at least in the author’s opinion) rather difficult to follow − the exposition is very terse, avoids technical details and leaves much of the work to the reader. The fact that the book is written in French does not make matters easier. Our goal is to overcome these inconveniences. As far as the organization of the paper is concerned, Chapter 2 is devoted to the Arzelà-Ascoli theorem. As we document, the classic version of this theorem (for C(X), X being a compact space) is very well- established in the literature, but the version for C0(X), X being a locally compact space, is almost non-existent. In our approach we employ the concept of one-point compactification X∞ of X. Following a brief description of this construct we lay out a full proof of the Arzelà-Ascoli theorem for C0(X). Chapter 3 commences with a short summary of the Haar measure. Subsequently, we introduce three Lp−properties: Lp−boundedness, Lp−equicontinuity and Lp−equivanishing. Next, Theorems 6, 7 and 5See [39], p. 53-54 for Weil’s argument (in french). 6See [11, 17, 18, 19, 20]. 2 9 demonstrate how these Lp−properties are “passed down” from F ⊂ Lp(G), G being a locally compact group, to F ⋆ φ ⊂ C0(G), where φ ∈ Cc(G). Chapter 4 is divided into two parts: the first one pivots around proving that F ⋆ φ is “not far away” (if we choose φ ∈ Cc(G) properly) from F (see Theorem 11). The subject of the second part is Young’s convolution inequality, which we generalize beyond what is currently known in the literature. Finally, Chapter 5 brings all the “pieces of the puzzle” from previous chapters together. In the climax of the paper (see Theorem 14) we provide an elegant and natural proof of the characterization of the relatively compact families in Lp(G). 2 Arzelà-Ascoli theorem and one-point compactification The classic version of the Arzelà-Ascoli theorem characterizes relatively compact families in C(X), the space of continuous functions on a compact space X (we assume that every topological space in this paper is Hausdorff). Such version of the result appears in many sources throughout the literature7, although some authors assume additionally that the space X is metric for “simplicity of the proof”8. Theorem 3. (classic version of the Arzelà-Ascoli theorem) Let X be a compact space. The family F ⊂ C(X) is relatively compact if and only if • F is pointwise bounded, i.e. for every x ∈ X there exists Mx > 0 such that ∀f∈F |f(x)| 6 Mx, • F is equicontinuous at every point, i.e. for every x ∈ X and ε> 0 there exists an open neighbourhood Ux of x such that ∀y∈Ux, |f(y) − f(x)| 6 ε. f∈F For our purposes, we need a more general result. We need a characterization of relatively compact families in C0(X), the space of continuous functions on a locally compact space X which vanish at infinity, i.e. for every ε> 0 there exists a compact set K in X (we denote this situation by K ⋐ X) such that ∀x∈X\K |f(x)| 6 ε. Looking through the literature, one usually stumbles across versions of the Arzelà-Ascoli theorem in which the supremum-norm topology is replaced with the compact-open topology9. Theorem 4. (Arzelà-Ascoli theorem for C0(X) with the compact-open topology) Let X be a locally compact space. The family F ⊂ C0(X) is relatively compact in the compact-open topology if and only if • F is pointwise bounded, • F is equicontinuous at every point. 7See Theorem 4.43 in [13], p. 137 or Corollary 10.49 in [25], p. 479 or Theorem A5 in [34], p. 394. 8See Theorem 4.25 in [2], p. 111 or Theorem 23.2 in [4], p. 99 or Theorem 6.3.1 in [9], p. 69 or Theorem 11.28 in [33], p. 245. 9See Theorem 3.4.20 in [10], p. 163 or Theorem 17 in [24], p. 233 or Theorem 47.1 in [27], p. 290. 3 The above result seems promising (to say the least), but it has a “splinter” in the form of the switch of topologies. Is there a version of the Arzelà-Ascoli theorem for C0(X) in which the supremum-norm topology is not replaced with compact-open topology (or any other topology for that matter)? The literature is surprisingly scarce in this respect. One possible reference is Exercise 17 on page 182 in John B. Conway’s “A Course in Functional Analysis”10. However, Conway does not provide a solution to his exercise, leaving this task to the reader. Another reference is the monograph “Integral Systems and Stability of Feedback Systems” by Constantin Corduneanu11. Unlike Conway, Corduneanu does provide a proof, but (in the author’s opinion) he does not provide an insight into why the theorem is true12. We aim to rectify this imperfection. For any locally compact space X, there exists a compact space X∞ such that • X is a subspace of X∞, • the closure of X is X∞, • X∞\X is a singleton or an empty set. 13 The space X∞ is called the one-point compactification of X . Its topology allows for a simple description − it consists of open sets in X (the topology of X) plus all sets of the form X∞\K, where K ⋐ X. If X happens to be compact itself, then X = X∞. Otherwise, X∞\X is a singleton, whose element is often denoted by ∞. A common example of the one-point compactification is C∞, which arises in the field of complex analysis as a natural domain for Möbius transformations. It turns out that C∞ is homeomorphic to a sphere S2, called the Riemann sphere, and the element ∞ may be regarded as the “north pole” of that sphere14. Theorem 5. (Arzelà-Ascoli theorem for C0(X) with the supremum-norm topology) Let X be a locally compact space. The family F ⊂ C0(X) is relatively compact in the supremum-norm topology if and only if • F is pointwise bounded, • F is equicontinuous at every point, • F is equivanishing, i.e.

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