Natural Proof of the Characterization of Relatively Compact Families in Lp

Home , Andrey Kolmogorov

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∀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. for every ε> 0 there exists K ⋐ X such that

∀x6∈K |f(x)| 6 ε. (1) f∈F

Proof. For every f ∈ C0(X) we define a function Ψf : X∞ −→ C such that Ψf |X = f and Ψf (∞)=0. Obviously, every Ψf is continuous at each element x ∈ X. Furthermore, functions Ψf are also continuous at ∞ : for a fixed ε > 0 we choose K ⋐ X such that (1) is satisfied and put U∞ := X∞\K, which is an open neighbourhood of ∞. Consequently, (1) corresponds to

∀x∈U∞ |Ψf (x) − Ψf (∞)| 6 ε,

10See [5]. 11See [6], p. 62. 12The idea of Cordeanu’s proof is basically to take a sequence and repeatedly choose some subsequences (using the assumptions) until we arrive at a convergent subsequence. This demonstrates the relative compactness of the family in question but still leaves the reader wondering “why is this theorem true?”. 13See Theorem 3.5.11 in [10], p. 169 or Theorem 29.1 in [27], p. 183 or Proposition 1.7.3 in [29], p. 37. 14See [14], p. 38 or [15], p. 10 or [36], p. 88-89.

4 which demonstrates that Ψf ∈ C(X∞) for every f ∈ C0(X). The crucial observation is that for every f ∈ C0(X) we have

sup |f(x)| = sup |Ψf (x)|, x∈X x∈X∞ so the function Ψ : C0(X) −→ C(X∞), given by Ψ(f) := Ψf , is an isometry. It is now clear that the following conditions are equivalent:

• F is relatively compact in C0(X),

• Ψ(F) is relatively compact in C(X∞),

• Ψ(F) is pointwise bounded and equicontinuous at every point x ∈ X∞ (by the classic Arzelà-Ascoli theorem).

Pointwise boundedness of Ψ(F) is equivalent to pointwise boundedness of F (since for every Ψf ∈ Ψ(F) we have Ψf (∞) = 0). Furthermore, equicontinuity of Ψ(F) at element x ∈ X is equivalent to equicontinuity of F at this element. Last but not least, the equicontinuity of Ψ(F) at ∞ means that for every ε > 0 there exists K ⋐ X such that

∀x∈X∞\K |F (x) − F (∞)| 6 ε, F ∈Ψ(F) which is equivalent to equivanishing of F. This concludes the proof.

3 Lp−properties

Throughout the whole paper, we work under the assumption that G is a locally compact (Hausdorﬀ) group. It turns out15 that any such group G admits a Haar measure µ, i.e. a nonzero, Borel measure which is

• ﬁnite on compact sets, • inner regular, i.e. for every open set U we have

µ(U) = sup {µ(K) : K ⊂ U, K − compact},

• outer regular, i.e. for every measurable set A we have

µ(A) = inf {µ(U) : A ⊂ U, U − open},

• left-invariant, i.e. for every x ∈ G and measurable set A we have µ(xA)= µ(A).

15See Chapter 1.3 in [8] or Chapter 2.2 in [12] or Section 15 in [23] or Chapter 2 in [39].

5 Haar measure is not determined uniquely, but is “unique up to a positive constant”, i.e. if µ1, µ2 are two (left) Haar measures on G then there exists a constant c> 0 such that µ1 = c · µ2. Furthermore, similarly to the left Haar measure, one can also prove the existence (and uniqueness up to positive constant) of the right Haar measure. In general, these two objects need not coincide on G - if they do, we call G unimodular. The family of unimodular groups contains all locally compact abelian groups as well as all compact groups. As far as the history is concerned, the construction of the Haar measure is commonly attributed to Alfréd Haar, although the contribution of Henri Cartan or André Weil deserves recognition as well16. For a detailed account of the subject we refer the Reader to Diestel’s and Spalsbury’s monograph “The joys of Haar measure”17. From this point onwards we assume that p > 1 and p′ is its Hölder conjugate. Our aim is to characterize relatively compact families of Lp(G) and to achieve this goal we introduce three concepts: ﬁrstly, we say that a family F ⊂ Lp(G) is Lp−bounded if there exists M > 0 such that

∀f∈F kfkp 6 M.

Secondly, a family F ⊂ Lp(G) is called Lp−equicontinuous if for every ε > 0 there exists an open neighbourhood Ue of the neutral element e ∈ G such that

∀f∈F sup kLxf − fkp 6 ε and sup kRxf − fkp 6 ε, x∈Ue x∈Ue

where Lxf(y) := f(xy) and Rxf(y) := f(yx) are the left and right shift operators, respectively. Thirdly, F ⊂ Lp(G) is said to be Lp−equivanishing if for every ε> 0 there exists K ⋐ G such that

p ∀f∈F |f(y)| dy 6 ε. ZG\K We now demonstrate that all three properties above (Lp−boundedness, Lp−equicontinuity and Lp−equivanishing) p are “inherited” when the family F ⊂ L (G) is convolved with a function φ ∈ Cc(G), i.e. a continuous function with compact support. As far as the notation is concerned, Cb(G) in the theorem below denotes the Banach space (with supremum-norm) of continuous and bounded functions on G.

p p b Theorem 6. Let F ⊂ L (G) be L −bounded. If φ ∈ Cc(G) then F ⋆ φ ⊂ C (G) is bounded. Proof. Let M > 0 be a Lp−bound on the family F. We divide the proof into two steps.

Step 1. Proving that f ⋆ φ is continuous for every f ∈F

Fix ε> 0, x∗ ∈ G and suppose that p> 1. By Proposition 2.41 in [12], p. 53 there exists a symmetric open neighbourhood Ue of the neutral element such that

1 ′ ′ p ε ∀ |φ ◦ ι(xy) − φ ◦ ι(y)|p dy 6 , (2) x∈Ue M ZG 16See [3, 22] or Chapter 2 in [39]. 17See [7].

6 −1 where ι : G −→ G stands for the inverse function, i.e. ι(x) := x . For x ∈ x∗Ue and f ∈F we have

−1 −1 |f ⋆ φ(x) − f ⋆ φ(x∗)| = f(y) · φ y x − f(y) · φ y x∗ dy G Z 1 ′ Hölder ineq. ′ p −1 −1 p 6 kfk p · φ y x − φ y x∗ dy G Z 1 ′ y7→x∗y ′ p −1 −1 −1 p 6 M · φ y x∗ x − φ y dy G Z 1 ′ ′ p (2) −1 p =M · |φ ◦ ι x x∗y − φ ◦ ι (y) | dy 6 ε. ZG This proves that F ⋆ φ is a family of continuous functions if p> 1. For p = 1 let us again ﬁx ε > 0 and x∗ ∈ G. By Lemma 1.3.6 in [8], p. 11 function φ is uniformly continuous, so there exists a symmetric open neighbourhood Ve of the neutral element such that

−1 −1 ε ∀x∈x∗Ve sup φ y x − φ y x∗ 6 . (3) y∈G M

Consequently, for x ∈ x∗Ve and f ∈F we have

−1 −1 |f ⋆ φ(x) − f ⋆ φ(x∗)| = f(y) · φ y x − f(y) · φ y x∗ dy ZG (3) ε 6 sup φ y−1x − φ y−1x · |f(y)| dy 6 ·kfk 6 ε. ∗ M 1 y∈G ZG This demonstrates that F ⋆ φ is a family of continuous functions if p =1.

Step 2. Proving that the family F ⋆ φ is bounded in supremum norm

Suppose that p> 1 and put K := ι(supp(φ)). We observe that

1 Hölder ineq. ′ p′ −1 −1 p ∀x∈G f(y) · φ y x dy 6 kfkp · φ y x dy f∈F G G Z 1 Z 1 ′ 6 M ·kφk · µ xK p = M ·kφk · µ(K ) p′ , ∞ ∞ where the second inequality stems from the fact that if

y−1x 6∈ supp(φ) ⇐⇒ y 6∈ xK,

then φ y−1x = 0. We conclude that F ⋆ φ ⊂ Cb(G) is bounded if p> 1. For p = 1, the reasoning is even simpler:

−1 ∀x∈G f(y) · φ y x dy 6 |f(y)| dy ·kφk∞ 6 M ·kφk∞. f ∈F ZG ZG

This concludes the proof.

7 Going oﬀ on a tangent for a brief moment, one may wonder whether µ(ι(supp(φ))) (which appears in the second step of the proof above) is equal to µ(supp(φ)). In general, this is not the case! However if (and only if) the group G is unimodular (left and right Haar measure coincide) then for every measurable set A we have µ(A)= µ(ι(A))18.

p p Theorem 7. Let F ⊂ L (G) be L −equicontinuous. If φ ∈ Cc(G), then F ⋆ φ is equicontinuous. p Proof. Firstly, suppose that p > 1, ε> 0, x∗ ∈ G and K := ι(supp(φ)). By L −equicontinuity of the family F, let Ue be a symmetric open neighbourhood of the neutral element such that ε ∀f∈F sup kLxf − fkp 6 1 . (4) ′ x∈Ue kφk∞ · µ(K) p Observe that for every x ∈ G and f ∈F we have

y7→xy f ⋆ φ(x)= f(y) · φ y−1x dy = f(xy) · φ y−1 dy. (5) ZG ZG Consequently, for every x ∈ x∗Ue and f ∈F we have

(5) −1 |f ⋆ φ(x) − f ⋆ φ(x∗)| 6 |f(xy) − f(x∗y)| · φ y dy G Z 1 1 ′ Hölder ineq. p ′ p p −1 p 6 |f(xy) − f(x∗y)| dy · |φ y | dy G G Z 1 Z p 1 p ′ 6 |f(xy) − f(x∗y)| dy ·kφk∞ · µ(K) p G Z 1 −1 p 1 ( ) y7→x∗ y p 4 −1 p′ = f xx∗ y − f(y) dy ·kφk∞ · µ(K) 6 ε, ZG which ends the proof if p> 1. 1 For p = 1 let us again ﬁx ε > 0 and x∗ ∈ G. By L −equicontinuity of the family F, let Ue be a symmetric open neighbourhood of the neutral element such that ε ∀f∈F sup kLxf − fk1 6 . (6) x∈Ue kφk∞

For every x ∈ x∗Ue and f ∈F, we obtain

8 Lemma 8. If K1 and K2 are compact subsets of G, then there exists a compact set D such that

∀x6∈D xK1 ∩ K2 = ∅. (7)

−1 Proof. We put D := K2 · K1 . It is obviously compact by the continuity of group operations and the compactness of K1 and K2. To prove that (7) is true, suppose that z ∈ xK1 ∩ K2 for some x 6∈ D. In particular z ∈ K2 and z = xy for some y ∈ K1. This means that

−1 −1 x = zy ∈ K2 · K1 = D, which is a contradiction. This concludes the proof.

p p Theorem 9. Let F ⊂ L (G) be L −equivanishing. If φ ∈ Cc(G) is such that φ(e) 6= 0 (e is the neutral element of G), then F ⋆ φ is equivanishing. Proof. Firstly, suppose that p> 1. We ﬁx ε> 0 and choose K ⋐ G such that

1 p p ε ∀f∈F |f| dµ 6 1 . (8) p′ ZG\K ! kφk∞ · µ ι(supp(φ)) Let us denote U := ι({φ 6=0}), which is open and relatively compact neighbourhood of the neutral element with U = ι(supp(φ)). By Lemma 8, there exists D ⋐ G such that

∀x6∈D xU ∩ K = ∅. (9)

For x 6∈ D and f ∈F we have

−1 |f ⋆ φ(x)| 6 f(y) · φ y x dy 6 kφk∞ · |f| dµ G xU Z 1 Z Hölder ineq. p 1 p ′ 6 kφk∞ · |f| dµ · µ(xU) p xU Z 1 p (9) 1 (8) p p′ 6 kφk∞ · |f| dµ · µ xU 6 ε, ZG\K ! which ends the proof if p> 1. For p = 1 let us again ﬁx ε> 0 and choose K ⋐ G such that ε ∀ |f| dµ 6 . (10) f∈F kφk ZG\K ∞ Furthermore, we choose U and D as previously. For every f ∈F and x 6∈ D we have

(9) (10) −1 |f ⋆ φ(x)| 6 f(y) · φ y x dy 6 kφk∞ · |f| dµ 6 kφk∞ · |f| dµ 6 ε. ZG ZxU ZG\K This concludes the proof.

9 4 Approximation theorem and Young’s convolution inequality

The ﬁrst part of this chapter is devoted to demonstrating that the family F ⋆φ (for a suitably chosen φ ∈ p 19 Cc(G)) approximates the family F in L −norm. To this end, we recall Minkowski’s integral inequality : Theorem 10. Let X, Y be σ−ﬁnite measure spaces, 1 6 p< ∞ and let F : X × Y −→ C be a measurable function. Then

1 1 p p p |F (x, y)| dy dx 6 |F (x, y)|p dx dy. (11) ZX ZY ZY ZX To a certain degree, the theorem below resembles Proposition 2.42 in Folland’s “A Course in Abstract Harmonic Analysis”20. However, Folland’s proposition focuses on a single function f and our approximation theorem deals with an Lp−equicontinuous family F ⊂ Lp(G).

p p Theorem 11. If F ⊂ L (G) is L −equicontinuous, then for every ε> 0 there exists a function φ ∈ Cc(G) such that ∀f∈F kf ⋆ φ − fkp 6 ε.

Proof. Fix ε> 0 and let Ue be the open neighbourhood of the neutral element such that

∀f∈F sup kRyf − fkp 6 ε. (12) y∈Ue

We ﬁx f ∈ F and choose fB to be a Borel-measurable function such that f = fB almost everywhere. Since G is a Tychonoﬀ space, we can pick φ ∈ Cc(G) such that • φ(e) 6=0, φ > 0,

• G φ ◦ ι dµ =1,

• Rsupp(φ ◦ ι) ⊂ Ue. For every x ∈ G we have

f ⋆ φ(x) − f(x)= f(y) · φ y−1x dy − f(x) · φ y−1 dy ZG ZG = f(xy) − f(x) · φ y−1 dy ZG −1 = fB(xy) − fB(x) · φ y dy. ZG We put −1 F (x, y) := fB(xy) − fB(x) · φ y ,

19See Theorem 6.19 in [13], p. 194. 20See [12], p. 53.

10 which is Borel-measurable as a composition of the following Borel-measurable functions:

F1 : (x, y) 7→ (x,y,y), −1 F2 : (x,y,z) 7→ (x,y,z ),

F3 : (x,y,z) 7→ (x, xy, φ(z)),

F4 : (x,y,z) 7→ (fB(x),fB (y),z) ,

F5 : (x,y,z) 7→ (y − x)z.

Observe that if we replace fB with f in F4, then F does not need to be Borel-measurable (or even measurable), since a composition of a measurable function with a continuous function need not be measurable. 21 Furthermore, since fB and φ are integrable with p−th power, then supp(fB) and supp(φ) are σ−compact . Hence, also the sets

(supp(fB) · supp(φ)) × ι(supp(φ)) and supp(fB) × ι(supp(φ)) are σ−compact. Following a series of logical implications:

−1 (x, y) ∈{F 6=0} =⇒ fB(xy) − fB(x) 6= 0 and φ y 6=0 −1 =⇒ (xy ∈ supp(fB) or x ∈ supp(fB)) and y ∈ supp(φ) −1 −1 =⇒ xy ∈ supp(fB) and y ∈ supp(φ) or x ∈ supp(fB) and y ∈ supp(φ) =⇒ (x, y) ∈ (supp(fB) · supp(φ)) × ι(supp(φ)) or (x, y) ∈ supp(fB) × ι(supp(φ)) we conclude that {F 6=0} is σ−compact. Finally, we are in position to apply Minkowski’s integral inequality:

1 p p −1 ∀f∈F kf ⋆ φ − fkp = kfB ⋆ φ − fBkp = fB(xy) − fB(x) · φ y dy dx ZG ZG 1 (11) p p −1 p 6 |fB(xy) − fB (x)| · φ y dx dy ZG ZG (12) −1 = kRyf − fkp · φ y dy 6 sup kRyf − fkp 6 ε. ZG y∈U This concludes the proof. We now turn to Young’s convolution inequality. Let ∆ represent the modular function associated with the Haar measure µ22. In [35] Quek and Yap prove Young’s convolution inequality for unimodular groups − these are locally compact groups in which ∆ = 1 (equivalently, left and right Haar measures coincide). Below we present our take on Young’s convolution inequality without the assumption of unimodularity.

21See Corollary 1.3.5 in [8], p. 10. 22See Chapter 1.4 in [8] or Chapter 2.4 in [12] or Section 15 in [23] or Chapter 3.3 in [31].

11 Theorem 12. Let p,q,r ∈ [1, ∞) be such that 1 1 1 = + − 1. (13) r p q

1 For functions f ∈ Lp(G) and g ∈ Lq(G), the convolution f⋆ ∆ p′ g exists almost everywhere. Moreover, 1 f ⋆ ∆ p′ g ∈ Lr(G) and we have 1 ′ kf ⋆ ∆ p g kr 6 kfkp ·kgkq. (14)

1 Proof. Observe that it suﬃces to prove (14), which will immediately establish that f ⋆ ∆ p′ g ∈ Lr(G) and consequently that the convolution exists almost everywhere. Firstly, we note a couple of useful equalities:

1 1 1 1 1 1 (13) + + = + 1 − + 1 − = 1, r q′ p′ r q p p (13) 1 1 − q′ = p 1 − q′ = p, (15) r q q (13) 1 1 − p′ = q 1 − p′ = q. r p With the aid of Hölder’s inequality, we perform the main calculation:

1 1 f ⋆ ∆ p′ g (x) = f(y) · ∆ p′ y−1x · g y−1x dy G Z q (1− ) p q p r 1 −1 r 1− −1 ′ −1 6 |f|(y) r · |g| y x · |f|(y)( r ) · |g| y x · ∆ p y x dy G Z 1 1 1 ′ ′ r p ′ q q ′ p p q 1− q (1− r )p 6 |f|(y) · |g| y−1x dy · |f|( r ) dµ · |g| y−1x · ∆ y−1x dy G G G Z 1 Z 1 Z 1 ′ ′ (15) q r q q p = |f|(y)p · |g| y−1x dy · |f|p dµ · |g| y−1x · ∆ y−1x dy G G G Z 1 Z Z 1 r p p′ y7→xy p −1 q q′ −1 q −1 = |f|(y) · |g| y x dy ·kfkp · |g| y · ∆ y dy G G Z 1 Z 1 r p p′ p −1 q q′ q = |f|(y) · |g| y x dy ·kfkp · |g| dµ G G Z 1 Z 1 r p q r p q p −1 q q′ p′ p q q′ p′ = |f|(y) · |g| y x dy ·kfkp ·kgkq = |f| ⋆ |g| (x) ·kfkp ·kgkq . ZG (16)

12 This leads to r 1 pr qr ′ p q q′ p′ f ⋆ ∆ p g (x) dx 6 |f| ⋆ |g| (x) dx ·kfkp ·kgkq G G Z Z pr qr ′ ′ p q q p = k|f| ⋆ |g| k1 ·kfkp ·kgkq pr qr pr qr ′ ′ p+ ′ q+ ′ p q q p q p 6 k|f| k1 · k|g| k1 ·kfkp ·kgkq = kfkp ·kgkq , where the second inequality follows from Theorem 1.6.2 in [8], p. 26. Taking the r-th root, we conclude that p p q qr r + q′ + p′ kf⋆gkr 6 kfkp ·kgkq = kfkp ·kgkq, which ends the proof.

5 Main theorem

Prior to demonstrating the main result of the paper we prove a simple lemma: p p p Lemma 13. Let F ⊂ L (G) be L −bounded and L −equicontinuous. If φ ∈ Cc(G) and K ⋐ G then F|K ⋆ φ is relatively compact in C0(G) and

∀f∈F supp f|K ⋆ φ ⊂ K · supp(φ).

p p Proof. Obviously, if F is L −bounded then so is F|K . Furthermore, F|K is trivially L −equivanishing, because it is supported in K. Last but not least, the Lp−equicontinuity of F implies the Lp−equicontinuity of F|K . By Theorems 6, 7 and 9 we conclude that F|K ⋆ φ is bounded, equicontinuous and equivanishing in C0(G). By Theorem 5 we establish that F|K ⋆ φ is relatively compact. For the last part of the theorem, observe that for every f ∈F we have

−1 ∀x∈G f|K ⋆ φ(x) = f(y) · φ y x dy. ZK The integral on the right-hand side is 0 for x 6∈ K · supp( φ), so

∀f∈F supp f|K ⋆ φ ⊂ K · supp(φ). This concludes the proof. At last, we have reached the climax of the paper. Let us recap how this main result relates to other compactness theorems known in the mathematical literature. To begin with, Theorem 14 generalizes the Kolmogorov-Riesz theorem23, as RN is a locally compact Hausdorﬀ group. Furthermore, we go to great lengths to render the proof below more transparent than the complicated Weil’s argument24. Last but not least, the ﬁnal part of our result generalizes the Sudakov theorem25.

23Apart from the papers [20], [26], [32], [38] mentioned in the Introduction, the proof of the Kolmogorov-Riesz theorem for RN can be found in [2], p. 111 (Theorem 4.26) or in [30], p. 21 (Theorem 1.3). 24See [39], p. 53-54 for comparison. 25See [21] or [37].

13 Theorem 14. A family F ⊂ Lp(G) is relatively compact if and only if • F is Lp−bounded, • F is Lp−equicontinuous, • F is Lp−equivanishing. Furthermore, suppose that the group G is such that for every open neighoburhood U of the neutral n element, there exists an element x ∈ U such that (x )n∈N is not contained in any compact set. Then the condition of Lp−boundedness is redundant. Proof. For the entire proof, which we divide into ﬁve steps, we ﬁx ε> 0.

Step 1. Relative compactness of F implies Lp−boundedness

This step is immediate, as a compact set is always bounded in a metric space.

Step 2. Relative compactness of F implies Lp−equicontinuity

N ε Let (fn)n=1 be an 3 −net for the family F. By Proposition 2.41 in [12], p. 53 for every n =1,...,N there exists an open neighbourhood Un of the neutral element such that ε ε sup kLxfn − fnkp 6 and sup kRxfn − fnkp 6 . (17) x∈Un 3 x∈Un 3

N Put U := n=1 Un, which is obviously an open set. Consequently, for every f ∈ F there exists n = 1,...,N such that T

sup kLxf − fkp 6 sup kLxf − Lxfnkp + sup kLxfn − fnkp + kfn − fkp x∈U x∈U x∈U (17) =2kfn − fkp + sup kLxfn − fnkp 6 ε. x∈U

p An analogous reasoning works for kRxf − fkp. We conclude that the family F is L −equicontinuous.

Step 3. Relative compactness of F implies Lp−equivanishing N ε ⋐ Let (fn)n=1 be an 2 −net for the family F. For every n =1,...,N there exists Kn G such that ε |f |p dµ 6 . (18) n 2 ZG\Kn N Put K := n=1 Kn, which is obviously a compact set. Consequently, for every f ∈ F there exists n =1,...,N such that S (18) ε ε |f|p dµ 6 |f − f |p dµ + |f |p dµ 6 + = ε. n n 2 2 ZG\K ZG\K ZG\K This proves that F is Lp−equivanishing.

14 Step 4. Lp−boundedness, Lp−equicontinuity and Lp−equivanishing imply relative compactness of F

Due to Theorem 11 we pick φ ∈ Cc(G) such that ε ∀ kf ⋆ φ − fk 6 . (19) f∈F p 4 By Lp−equivanishing of F, there exists K ⋐ G such that ε ∀f∈F kf − f|K kp 6 1 . (20) − ′ 4 ·k∆ p φk1

Due to Theorem 13 we know that F|K ⋆ φ is relatively compact in C0(G) and that

∀f∈F supp(f|K ⋆ φ) ⊂ D,

where D := K · supp(φ). By relative compactness of F|K ⋆ φ there exists a ﬁnite sequence of functions N (gn)n=1 ⊂ Cc(G) such that for every f ∈F there exists n =1,...,N such that ε f ⋆ φ g 6 1 . |K − n ∞ (21) 4 · µ(D) p

Due to inner regularity of µ there exists an open set V ⊃ D such that

ε p µ(V \D) 6 . (22) 4 · max kg k n=1,...,N n ∞ Since every locally compact group is normal26, by Urysohn’s lemma27 there exists a function u : G −→ [0, 1] such that

u|D = 1 and u|G\V =0. (23)

Finally, for every f ∈F there exists n =1,...,N such that

ε 1 1 (21), (22) 6 + kf| ⋆ φ − g k · µ(D) p + kg k · µ(V \D) p 6 ε. 2 K n ∞ n ∞

N This demonstrates that (gn · u)n=1 is an ε−net for F. Thus F is relatively compact. Step 5. Sudakov’s part

26See Theorem 8.13 in [23], p. 76. 27See Theorem 1.5.11 in [10], p. 41 or Lemma 4 in [24], p. 115 or Theorem 33.1 in [27], p. 207 or Theorem 1.5.6 in [29], p. 24.

15 We will now prove that Lp−boundedness follows from Lp−equicontinuity and Lp−equivanishing under the assumption that for every open neighoburhood U of the neutral element, there exists an element x ∈ U n p such that (x )n∈N is not contained in any compact set. By L −equicontinuity of F there exists an open neighbourhood Ue of the neutral element such that

1 p p ∀x∈Ue |Lxf − f| dµ 6 1. (24) f∈F ZG Furthermore, due to Lp−equivanishing there exists K ⋐ G such that

1 p p ∀f∈F |f| dµ 6 1. (25) ZG\K ! n Let x∗ ∈ Ue be an element such that (x∗ )n∈N is not contained in any compact set. For every f ∈ F we have

1 1 1 p p p p p p |f| dµ 6 |Lx∗ f − f| dµ + |Lx∗ f| dµ K K K Z Z 1 Z 1 (24) p p p p 6 1+ |f(x∗y)| · 1K (y) dy =1+ |f| dµ . ZG Zx∗K Using an inductive reasoning we have

1 1 p p p p ∀n∈N |f| dµ 6 n + |f| dµ . (26) n ZK Zx∗ K ! N N Observe that there exists N ∈ such that x∗ K ∩ K = ∅, since otherwise we would have (xn)n∈N ⊂ K · K−1, contrary to our assumption. Finally, we have

1 1 1 p p p (25) p p p kfkp 6 |f| dµ + |f| dµ 6 N + |f| dµ +1 6 N +2, N ZK ZG\K ! Zx∗ K ! which ends the proof. Corollary 15. A family F ⊂ Lp(R) is relatively compact if and only if • F is Lp−equicontinuous, • F is Lp−equivanishing.

Proof. The corollary follows from Theorem 14 and the fact that if U0 is an open neighbourhood of the neutral element (namely 0) and if x ∈ U0, x 6= 0 then the sequence (n · x)n∈N is not contained in any compact subset of R. As a ﬁnal remark let us consider the space ℓp(Z). Its relatively compact subsets are characterized as follows28: 28See [20].

16 Theorem 16. A family F ⊂ ℓp(Z) is relatively compact if and only if • F is ℓp−bounded, • F is ℓp−equivanishing, i.e. for every ε> 0 there exists N ∈ N such that

p ∀x∈F |xn| 6 ε. |nX|>N

Firstly, let us observe that Theorem 16 does not mention “ℓp−equicontinuity” − this is because in ℓp(Z) every family is ℓp−equicontinuous. Furthermore, we argue that ℓp−boundedness is not a redundant condition in characterizing relatively compact families in ℓp(Z). This does not contradict the ﬁnal part of Theorem 14, because the singleton {0} is an open neighbourhood of the neutral element (namely 0) and the sequence (n · 0)n∈N (the only possible sequence we can construct from the single element of {0}) is contained in the compact set {0}. p Let x ∈ ℓ (Z) be such that x0 = 1 and xn = 0 for n ∈ Z\{0}. Consider the family F = (n · x)n∈N. Obviously, the family F is ℓp−equivanishing, but it is not ℓp−bounded. Furthermore, it is not relatively compact which proves that ℓp−boundedness is indeed indispensible for the Theorem 16 to work.

Acknowldegements

I would like to express my gratitude towards Wojciech Kryszewski, whose insightful questions forced me to repeatedly rethink the ideas that I have been working on. It is a privilege to have such a great source of constructive criticism. I also wish to thank Robert Stańczy for the reference to the Arzelà-Ascoli theorem for C0(X) in the monograph “Integral Equations and Stability of Feedback Systems” by Constantin Corduneanu.

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