Matrix weighted Kolmogorov-Riesz's compactness theorem
aa r X i v : . [ m a t h . C A ] F e b MATRIX WEIGHTED KOLMOGOROV-RIESZ’S COMPACTNESSTHEOREM
SHENYU LIU, DONGYONG YANG AND CIQIANG ZHUO
Abstract.
In this paper, several versions of the Kolmogorov-Riesz compactness theo-rem in weighted Lebesgue spaces with matrix weights are obtained. In particular, whenthe matrix weight W is in the known A p class, a characterization of totally boundedsubsets in L p ( W ) with p ∈ (1 , ∞ ) is established. Introduction
In this paper, we investigate totally bounded sets in matrix weighted Lebesgue spaces,from which one can obtain corresponding compactness criteria via the Hausdorff criterionfor compactness, that is, a set is precompact if and only if it is complete and totallybounded.In classical Lebesgue spaces L p , the characterization of precompact sets was givenby the celebrated Kolmogorov-Riesz theorem (see [22]) which was first discovered byKolmogorov [27] in L p ([0 , p ∈ (1 , ∞ ). Subsequently, Tamarkin [36] extended theresult to the case in which the underlying space can be unbounded, with an additionalcondition related to the behaviour at infinity. Tulajkov [39] showed that Tamarkin’sresult was also true when p = 1. At the same time, Riesz [32] independently proved asimilar result. Since then, compactness criteria of subsets in Lebesgue spaces have beenstudied and applied in various settings, e.g. see [35, 42] for some improvements andapplications on Kolmogorov-Riesz’s theorem, and see [31, 19, 20, 2, 3, 1] for a series ofworks on compactness criteria in variable exponent function spaces.In particular, Tsuji [38] showed that Kolmogorov-Riesz’s theorem is true in L p ( R ) for p ∈ (0 ,
1) and his method has been applied by many authors, e.g. [43, 21, 7]. Moreover,based on the Arzel´a-Ascoli theorem (see [17]), the authors in [34, 17] established com-pactness criteria in Lebesgue-Bochner spaces. In addition, the authors in [26, 28] appliedthe so-called Lebesgue-Vitali’s theorem and established compactness criteria in L ( m ),the space of all Lebesgue measurable functions on R n that are finite almost everywhere,where m denotes the Lebesgue measure.Recently, Hanche-Olsen–Holden [22] seminally showed that both the Arzel´a-Ascoli the-orem and Kolmogorov-Riesz’s theorem are consequences of a simple lemma on compact-ness in metric spaces via a finite dimension argument. Inspired by the method used in[22], Clop–Cruz [9] first gave a compactness criterion in scalar weighted Lebesgue spaces L p ( ω ) for p ∈ (1 , ∞ ) with a weight ω ∈ A p . Their result was then improved by Guo-Zhaoin [21], in which they gave the following compactness criterion in L p ( ω ) for p ∈ (0 , ∞ )with ω ∈ L ( R n ). Date : February 3, 2021.2020
Mathematics Subject Classification.
Key words and phrases.
Kolmogorov-Riesz theorem, matrix weights, totally bounded, metric measurespaces, variable exponent Lebesgue spaces.
Theorem A.
Let < p < ∞ and ω ∈ L ( R n ) be a nonnegative function. A subset F of L p ( ω ) is totally bounded if the following conditions hold:(a) F is bounded, i.e. sup f ∈F k f k L p ( ω ) < ∞ ; (b) F uniformly vanishes at infinity, that is, lim R →∞ sup f ∈F k f χ B c (0 ,R ) k L p ( ω ) = 0; (c) F is equicontinuous, that is, lim r → sup f ∈F sup y ∈ B (0 ,r ) k τ y f − f k L p ( ω ) = 0 . Here, τ y denotes the translation operator: τ y f ( x ) := f ( x − y ) . There is a natural question whether Theorem A can be extended to the setting ofmatrix weights. As a natural vector-valued generalization of scalar Muckenhoupt A p weights, the theory of matrix weights was first introduced by Bloom [4, 5] in 1981.Then the theory was pushed forward through the seminal work of Nazarov-Treil-Volberg[30, 40, 37], Christ-Goldberg [8, 18], and Frazier-Roudenko [33, 15] in the late 1990s, thatwas arose from problems in the theory of stationary processes, the theory of Toeplitzoperators and multivariable elliptic PDEs. From then on, harmonic analysis with matrixweights have been considered by many authors in various directions. For the references,we refer to [11, 12] for recent developments on matrix weights, [29, 24, 25] for the matrix A conjecture related to the sharp norm estimates for singular operators, and [13, 14] forsome applications of matrix weights.Matrix weights share many properties with scalar weights, for example, the definitionof matrix A p weights due to Frazier-Roudenko [33, 15] seems to be an intuitive extensionof the definition of scalar A p weights and the Hilbert transform is bounded on L p ( W ) ifand only if W ∈ A p . However, due to the non-commutativity in the matricial setting,many techniques of the classical harmonic analysis fail to generalize to the case of vector-valued functions with matrix weights, so the vector-valued case cannot be easily reducedto the scalar case. For example, in the setting of matrix weights, a suitable theory ofweak-type spaces L p, ∞ ( W ) is unknown.In this paper, we obtain several generalizations of Theorem A in weighted Lebesguespaces with matrix weights. To be precise, we first obtain a Kolmogorov-Riesz theoremin the weighted variable Lebesgue space L p ( · ) ( ρ ) on ( R n , | · | , m ), where ρ = { ρ x } x ∈ R n isa family of norms on C d , and another version of compactness criterion in L p ( W ) when p ∈ (0 , ∞ ) and W is a matrix weight on R n . We also establish a Kolmogorov-Riesztheorem in L p ( ρ, µ ) with p ∈ [1 , ∞ ) on metric measure spaces ( X, d, µ ), and obtain anequivalent characterization of precompact subsets in L p ( W ) for p ∈ (1 , ∞ ) and W in A p class on R n as an application.We would like to emphasize that due to the special structure of matrix weightedLebesgue spaces, neither matrix weighted Lebesgue spaces L p ( W ) nor weighted Lebesguespaces L p ( ρ ) are in the framework of (quasi-)Banach function spaces in [20, 6, 21] orLebesgue-Bochner spaces in [34, 17]. For example, in the case of vector-valued functionswith matrix weights, much of the ability to compare objects and dominate one by anotheris lost. Moreover, for a scalar weight ω , we have the fact that a function f ∈ L p ( ω ) ifand only if | f | ∈ L p ( ω ). Unfortunately, it is not true for matrix weights.Based on these facts, different from the classical case, the available methods to proveKolmogorov-Riesz’s theorem seems to be not applicable in the setting of matrix weights. ATRIX WEIGHTED KOLMOGOROV-RIESZ’S COMPACTNESS THEOREM 3
And we use a finite dimension argument without using the key lemma in [22] to obtaincompactness criteria in matrix weighted Lebesgue spaces on ( R n , | · | , m ).The paper is organized as follows. In Section 2, we recall some basic notations and factsrelated to matrix weights. In particular, we recall the so-called John ellipsoid theoremwhich shows the existence of a positive-definite self-adjoint matrix for a given norm ρ on C d ; see Lemma 2.2 below.Section 3 is devoted to the study of totally bounded sets in matrix weighted Lebesguespaces on ( R n , | · | , m ). For a given family of norm ρ = { ρ x } x ∈ R n on C d , we first apply theJohn ellipsoid theorem and establish a version of Kolmogorov-Riesz’s theorem in L p ( · ) ( ρ )for an exponent function p ( · ). When p ∈ (0 , ∞ ) and W is a matrix weight on R n whichis not necessarily invertible, we also obtain a Kolmogorov-Riesz theorem in L p ( W ) byfollowing some idea from [13]. As an application, a compactness criterion in degenerateSobolev spaces with matrix weights is given.In Section 4, let ( X, d, µ ) be a proper metric measure space such that µ is continuouswith respect to the metric d and ρ = { ρ x } x ∈ X be a family of norms on C d . We presenta Kolmogorov-Riesz theorem in weighted Lebesgue spaces L p ( ρ, µ ) with p ∈ [1 , ∞ ) on( X, d, µ ) in terms of the average operator, and apply to L p ( W, µ ) when W is an invertiblematrix weight on R n and µ is continuous with respect to | · | . We would like to mentionthat our method to prove Theorem 3.2 (see also [22, Theorem 5] and [21, Theorem 3.1])relies on the translation invariance of the Euclidean metric and the Lebesgue measure,which fails on metric measure spaces.In Section 5, based on the result obtained in Section 4, when the matrix weight W isin the known A p class on R n in [33], we further obtain an equivalent characterization ofcompact subsets in L p ( W ) for p ∈ (1 , ∞ ).Throughout this paper, we will use the following notations. We always use ω ( · ) todenote a scalar weight while W ( · ) to denote a matrix weight. Given two values A and B , we will write A . B if there exists a positive constant c , independent of appropriatequantities involved in A and B , such that A ≤ cB . We write A ≈ B if A . B and B . A . We will use p ′ to denote the conjugate exponent of p when p ∈ (1 , ∞ ). For agiven set E , χ E means the characteristic function of E . Additionally, unless otherwisenoted, ( R n , | · | , m ) is the underlying measure space.2. Preliminaries
In this section, we recall some basic notations and facts about matrix weights; see[18, 13] and the references therein.Let M d denote the set of all complex-valued, d × d matrices. A matrix function on R n is a map W : R n → M d . We say that it is measurable if each component of W isa measurable function, and invertible if det W ( x ) = 0 a.e. and so W − exists. Let S d be the set of all those A ∈ M d that are self-adjoint and non-negative-definite. For each A ∈ S d , A has d non-negative real-valued eigenvalues λ i , 1 ≤ i ≤ d , and the norm of A is defined as the operator norm k A k op := sup v ∈ C d , | v | =1 | A v | = max i λ i . Moreover, there exists a unitary matrix U such that U H AU is diagonal, where U H denotesthe conjugate transpose matrix of U . We denote a diagonal matrix by D ( λ , · · · , λ d ) = D ( λ i ). For every s >
0, we define A s := U D ( λ si ) U H . Furthermore, if A is positive-definite, we set A − s := U D ( λ − si ) U H . SHENYU LIU, DONGYONG YANG AND CIQIANG ZHUO
The following technical lemma is from [13, Lemma 3.1].
Lemma 2.1.
Given a measurable matrix function W : R n → S d , there exists a d × d measurable matrix function U defined on R n such that U H ( x ) W ( x ) U ( x ) is diagonal, and U ( x ) is unitary for every x ∈ R n . Based on Lemma 2.1, it is easy to see that for any measurable matrix function W : R n → S d , W s is a measurable matrix function satisfying that, for any x ∈ R n , k W s ( x ) k op = max i λ si ( x ) . (2.1)If W is invertible, W − s is a measurable matrix function satisfying that, for any x ∈ R n , k W − s ( x ) k − op = min i λ si ( x ) . (2.2)By a matrix weight on R n we mean a measurable matrix function W : R n → S d suchthat k W k op ∈ L ( R n ). Equivalently, each eigenvalue function λ i ∈ L ( R n ) , ≤ i ≤ d .Define the matrix weighted Lebesgue space L p ( W ) for p ∈ (0 , ∞ ) to be the set of allmeasurable vector-valued functions f := ( f , · · · , f d ) T : R n → C d such that k f k pL p ( W ) := Z R n (cid:12)(cid:12) W p ( x ) f ( x ) (cid:12)(cid:12) p dx < ∞ . In many cases, it is more convenient to characterize matrix weighted Lebesgue spacesin the following language; see [40]. Let ρ := { ρ x } x ∈ R n be a family of norms on C d , wherefor each x ∈ R n , ρ x : C d → R + := [0 , ∞ ). Define the weighted Lebesgue space L p ( ρ ) for p ∈ (0 , ∞ ) to be the set of all measurable vector-valued functions f : R n → C d such that k f k pL p ( ρ ) := Z R n [ ρ x ( f ( x ))] p dx < ∞ , where we always assume that ρ x ( f ( x )) is a measurable function on R n for any measurablevector-valued function f .For any given invertible matrix weight W , one can reduce L p ( ρ ) to L p ( W ) by setting ρ x ( · ) := | W p ( x ) · | . The following so-called John ellipsoid theorem (see [18, Proposition1.2]) shows that the two matrix weighted Lebesgue spaces above actually coincide. Lemma 2.2.
Given a norm ρ on C d , there exists a positive-definite self-adjoint matrix W such that ρ ( v ) ≤ | W ( v ) | ≤ d ρ ( v ) , ∀ v ∈ C d . (2.3)We now recall the definition of matrix A p weights due to Frazier-Roudenko [33, 15];see also [30] for another definition of matrix A p weights when p > Definition 2.3.
Let W be an invertible matrix weight.(i) When p ∈ (1 , ∞ ), we say W ∈ A p if k W − k p ′ p op ∈ L ( R n ) and[ W ] A p := sup Q | Q | Z Q (cid:18) | Q | Z Q (cid:13)(cid:13) W p ( x ) W − p ( y ) (cid:13)(cid:13) p ′ op dy (cid:19) pp ′ dx < ∞ , where the supremum is taken over all cubes Q ⊂ R n .(ii) When p ∈ (0 , W ∈ A p if k W − k op ∈ L ( R n ) and[ W ] A p := sup Q esssup x ∈ Q | Q | Z Q (cid:13)(cid:13) W p ( y ) W − p ( x ) (cid:13)(cid:13) pop dy < ∞ , where the first supremum is taken over all cubes Q ⊂ R n . ATRIX WEIGHTED KOLMOGOROV-RIESZ’S COMPACTNESS THEOREM 5
We would like to mention that when p ≥ d = 1 and W ( x ) = ω ( x ) is a scalarweight, the matrix A p condition is the Muckenhoupt A p condition. Moreover, we havethe following lemma due to [13, Lemma 4.5]. Lemma 2.4.
Let < p < ∞ . If W ∈ A p , then k W k op and k W − k − op are scalar A p weights. Next we recall a variant of the maximal operator introduced by Christ-Goldberg in[8, 18]. The Christ-Goldberg maximal operator M ω is defined as M ω f ( x ) := sup B ∋ x | B | Z B (cid:12)(cid:12) W p ( x ) W − p ( y ) f ( y ) (cid:12)(cid:12) dy, where the supremum is taken over all balls in R n containing x . They obtained thefollowing strong-type estimate for the Christ-Goldberg maximal operator. Lemma 2.5.
Let < p < ∞ . If W ∈ A p , then there exists δ > such that when q ∈ { q > | p − q | < δ } , k M ω f k L q ( R n ) . k f k L q ( R n , C d ) , ∀ f ∈ L q ( R n , C d ) , where the implicit constant depends only on q and k f k L q ( R n , C d ) := (cid:18) Z R n | f ( x ) | q dx (cid:19) q . Compactness criteria on R n This section is devoted to the study of Kolmogorov-Riesz’s theorem in matrix weightedLebesgue spaces on R n . In [19, Theorem 5], G´orka-Macios established a compactnesscriterion in variable exponent Lebesgue spaces L p ( · ) ( R n ). The first main result of thissection is to obtain a generalization of [19, Theorem 5] to L p ( · ) ( ρ ). Before that, we recallsome basic notations and results about variable exponent Lebesgue spaces; see [10].Let P ( R n ) be the set of all measurable functions p ( · ) : R n → [1 , ∞ ]. The elements of P ( R n ) are called exponent functions. Given an exponent function p ( · ) ∈ P ( R n ), we put p + := esssup x ∈ R n p ( x ) , p − := essinf x ∈ R n p ( x ) . We assume that exponent functions p ( · ) are bounded, i.e. p + < ∞ . Define the weightedvariable Lebesgue space L p ( · ) ( ρ ) to be the set of all measurable vector-valued functions f : R n → C d such that the modular Z R n [ ρ x ( f ( x ))] p ( x ) dx < ∞ , equipped with the Luxemburg norm k f k L p ( · ) ( ρ ) := inf (cid:26) λ > Z R n (cid:20) ρ x ( f ( x )) λ (cid:21) p ( x ) dx ≤ (cid:27) . According to the above definition and the convexity of the modular, we obtain thefollowing useful results on the relationship between the modular and the norm.
Lemma 3.1.
Let p + < ∞ , < λ ≤ , and f ∈ L p ( · ) ( ρ ) . Then the following statementsare true: ( a ) If k f k L p ( · ) ( ρ ) ≤ , then R R n [ ρ x ( f ( x ))] p ( x ) dx ≤ k f k L p ( · ) ( ρ ) ;( b ) If k f k L p ( · ) ( ρ ) > , then R R n [ ρ x ( f ( x ))] p ( x ) dx ≥ k f k L p ( · ) ( ρ ) ; SHENYU LIU, DONGYONG YANG AND CIQIANG ZHUO ( c ) k f k L p ( · ) ( ρ ) ≤ R R n [ ρ x ( f ( x ))] p ( x ) dx + 1;( d ) If R R n [ ρ x ( f ( x ))] p ( x ) dx ≤ λ p + , then k f k L p ( · ) ( ρ ) ≤ λ .Proof. ( a )–( c ) hold by adapting the arguments in [10, Corollary 2.22]. To prove ( d ),noting that when p + < ∞ and 0 < λ ≤
1, we have Z R n (cid:20) ρ x ( f ( x )) λ (cid:21) p ( x ) dx ≤ Z R n [ ρ x ( f ( x ))] p ( x ) λ − p + dx ≤ , which implies k f k L p ( · ) ( ρ ) ≤ λ . (cid:3) Based on Lemma 3.1, we now present a Kolmogorov-Riesz theorem in L p ( · ) ( ρ ). Theorem 3.2.
Let p ( · ) ∈ P ( R n ) , p + < ∞ and ρ := { ρ x } x ∈ R n be a family of norms on C d such that W x is an invertible matrix weight satisfying (2.3) for every x ∈ R n and k W x k p + op ∈ L ( R n ) . A subset F ⊂ L p ( · ) ( ρ ) is totally bounded if the following conditionsare valid: ( a ) F is bounded in the sense of the modular, that is, sup f ∈F Z R n [ ρ x ( f ( x ))] p ( x ) dx < ∞ ;( b ) F uniformly vanishes at infinity, that is, lim R →∞ sup f ∈F Z B c (0 ,R ) [ ρ x ( f ( x ))] p ( x ) dx = 0;( c ) F is equicontinuous, that is, lim r → sup f ∈F sup y ∈ B (0 ,r ) Z R n [ ρ x ( τ y f ( x ) − f ( x ))] p ( x ) dx = 0 . Proof.
Assume that
F ⊂ L p ( · ) ( ρ ) satisfies ( a )–( c ). Given ǫ > F , it suffices to find a finite ǫ -net of F . Denote by R i := [ − i , i ) n for i ∈ Z . Then by condition ( b ), there exists a positive integer m large enough such thatsup f ∈F Z R n (cid:2) ρ x (cid:0) f ( x ) − f ( x ) χ R m ( x ) (cid:1)(cid:3) p ( x ) dx < ǫ. (3.1)Moreover, by condition ( c ), there exists an integer t such thatsup f ∈F sup y ∈ R t Z R n (cid:2) ρ x (cid:0) f ( x − y ) − f ( x ) (cid:1)(cid:3) p ( x ) dx < ǫ. (3.2)Let Q i , i ∈ Z , be the family of dyadic cubes in R n , open on the right, whose verticesare adjacent points of the lattice (2 i Z ) n . For each i ∈ Z , the cubes in Q i are eitherdisjoint or coincide. Thus there exists a sequence { Q j } Nj =1 of disjoint cubes in Q t suchthat R m = N S j =1 Q j , where N = 2 ( m +1 − t ) n is a positive integer.For any f ∈ F and x ∈ R n , defineΦ( f )( x ) := ( f Q j := | Q j | R Q j f ( y ) dy, x ∈ Q j , j = 1 , · · · , N, , otherwise . ATRIX WEIGHTED KOLMOGOROV-RIESZ’S COMPACTNESS THEOREM 7
Then for every fixed x ∈ R n , by Lemma 2.2, there exists a positive-definite self-adjointmatrix W x such that for each j , ρ x (cid:0)(cid:0) f ( x ) − f Q j (cid:1) χ Q j ( x ) (cid:1) ≤ (cid:12)(cid:12) W x (cid:0) f ( x ) − f Q j (cid:1) χ Q j ( x ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) | Q j | Z Q j W x (cid:0) f ( x ) − f ( y ) (cid:1) dyχ Q j ( x ) (cid:12)(cid:12)(cid:12)(cid:12) . | Q j | Z Q j (cid:12)(cid:12) W x (cid:0) f ( x ) − f ( y ) (cid:1)(cid:12)(cid:12) dyχ Q j ( x ) . | Q j | Z Q j ρ x (cid:0) f ( x ) − f ( y ) (cid:1) dyχ Q j ( x ) , where the implicit constant depends only on d . Then from the choices of { Q j } Nj =1 , theJensen inequality, the Fubini theorem and (3.2), it follows that Z R n (cid:2) ρ x (cid:0) f ( x ) χ R m ( x ) − Φ( f )( x ) (cid:1)(cid:3) p ( x ) dx . N X j =1 Z Q j (cid:12)(cid:12)(cid:12)(cid:12) | Q j | Z Q j ρ x (cid:0) f ( x ) − f ( y ) (cid:1) dy (cid:12)(cid:12)(cid:12)(cid:12) p ( x ) dx . N X j =1 | Q j | Z Q j Z Q j (cid:2) ρ x (cid:0) f ( x ) − f ( y ) (cid:1)(cid:3) p ( x ) dx dy ≈ − nt N X j =1 Z Q j Z Q j (cid:2) ρ x (cid:0) f ( x ) − f ( y ) (cid:1)(cid:3) p ( x ) dy dx ≈ − nt N X j =1 Z Q j Z x − Q j (cid:2) ρ x (cid:0) f ( x ) − f ( x − y ) (cid:1)(cid:3) p ( x ) dy dx . − nt Z R n Z R t (cid:2) ρ x (cid:0) f ( x ) − f ( x − y ) (cid:1)(cid:3) p ( x ) dy dx ≈ − nt Z R t Z R n (cid:2) ρ x (cid:0) f ( x ) − f ( x − y ) (cid:1)(cid:3) p ( x ) dx dy . − nt | R t | sup y ∈ R t Z R n (cid:2) ρ x (cid:0) f ( x ) − f ( x − y ) (cid:1)(cid:3) p ( x ) dx . n ǫ, (3.3)where we use the fact that x − Q j := { x − y : y ∈ Q j } ⊂ R t when x ∈ Q j . Note that Z R n (cid:2) ρ x (cid:0) f ( x ) − Φ( f )( x ) (cid:1)(cid:3) p ( x ) dx = (cid:20) Z R n \ R m + Z R m (cid:21)(cid:2) ρ x (cid:0) f ( x ) − Φ( f )( x ) (cid:1)(cid:3) p ( x ) dx = Z R n (cid:2) ρ x (cid:0) f ( x ) − f ( x ) χ R m ( x ) (cid:1)(cid:3) p ( x ) dx + Z R n (cid:2) ρ x (cid:0) f ( x ) χ R m ( x ) − Φ( f )( x ) (cid:1)(cid:3) p ( x ) dx. This via (3.1) and (3.3) implies thatsup f ∈F Z R n (cid:2) ρ x (cid:0) f ( x ) − Φ( f )( x ) (cid:1)(cid:3) p ( x ) dx . ǫ, (3.4) SHENYU LIU, DONGYONG YANG AND CIQIANG ZHUO where the implicit constant depends only on n and d . Since p + < ∞ , then from (3.4)and Lemma 3.1, it suffices to show that Φ( F ) is totally bounded in L p ( · ) ( ρ ).Note that by condition ( a ), we havesup f ∈F Z R n (cid:2) ρ x (cid:0) Φ( f )( x ) (cid:1)(cid:3) p ( x ) dx ≤ p + − (cid:18) sup f ∈F Z R n (cid:2) ρ x (cid:0) f ( x ) − Φ( f )( x ) (cid:1)(cid:3) p ( x ) dx + sup f ∈F Z R n (cid:2) ρ x (cid:0) f ( x ) (cid:1)(cid:3) p ( x ) dx (cid:19) < ∞ . Then by [10, Remark 2.10], it follows that for any f ∈ F , | W x Φ( f )( x ) | ≤ d ρ x (cid:0) Φ( f )( x ) (cid:1) < ∞ a.e. x ∈ R n . Since W x is positive-definite for every x ∈ R n , we obtain | Φ( f )( x ) | < ∞ a.e. x ∈ R n , which implies | f Q j | < ∞ , j = 1 , · · · , N. From this and k W x k p + op ∈ L ( R n ), we see thatΦ is a map from F to B , a finite dimensional Banach subspace of L p ( · ) ( ρ ). Notice thatΦ( F ) ⊂ B is bounded, and hence is totally bounded. The proof of Theorem 3.2 iscomplete. (cid:3) As a corollary of Theorem 3.2, by taking p ( · ) ≡ p ∈ [1 , ∞ ), we have the followingKolmogorov-Riesz theorem in L p ( ρ ) defined in Section 2. Corollary 3.3.
Let ≤ p < ∞ , ρ := { ρ x } x ∈ R n be a family of norms on C d such that W x is an invertible matrix weight satisfying (2.3) for every x ∈ R n and k W x k pop ∈ L ( R n ) .A subset F of L p ( ρ ) is totally bounded if the following conditions hold: ( a ) F is bounded, i.e. sup f ∈F k f k L p ( ρ ) < ∞ ;( b ) F uniformly vanishes at infinity, that is, lim R →∞ sup f ∈F k f χ B c (0 ,R ) k L p ( ρ ) = 0;( c ) F is equicontinuous, that is, lim r → sup f ∈F sup y ∈ B (0 ,r ) k τ y f − f k L p ( ρ ) = 0 . As an application of Corollary 3.3, by setting ρ x ( · ) := | W p ( x ) · | and (2.1), we havethe following compactness criterion in L p ( W ) for p ∈ [1 , ∞ ). Corollary 3.4.
Let ≤ p < ∞ , W be an invertible matrix weight. A subset F of L p ( W ) is totally bounded if the following conditions hold: ( a ) F is bounded, i.e. sup f ∈F k f k L p ( W ) < ∞ ;( b ) F uniformly vanishes at infinity, that is, lim R →∞ sup f ∈F k f χ B c (0 ,R ) k L p ( W ) = 0;( c ) F is equicontinuous, that is, lim r → sup f ∈F sup y ∈ B (0 ,r ) k τ y f − f k L p ( W ) = 0 . ATRIX WEIGHTED KOLMOGOROV-RIESZ’S COMPACTNESS THEOREM 9
Remark 3.5.
It is worth noting that compared with the compactness criteria in [9, 43, 7],there is no additional assumption on W . Based on this, from the fact that matrix weightedLebesgue spaces are not translation invariant, on which the translation operator τ y is notcontinuous, it follows that Corollary 3.4 is a strong sufficient condition for precompactnessin L p ( W ).When p ∈ (0 , ∞ ) and W is a matrix weight which is not necessarily invertible, wealso have the following characterization of totally bounded sets in L p ( W ), which extendsTheorem A to the setting of matrix weights. Theorem 3.6.
Let < p < ∞ , W be a matrix weight. A subset F of L p ( W ) is totallybounded if it satisfies the conditions ( a ) , ( b ) in Corollary 3.4 and ( c ∗ ) F is equicontinuous in the sense that lim r → sup f ∈F sup y ∈ B (0 ,r ) (cid:18) Z R n (cid:12)(cid:12)(cid:12) D p ( x ) (cid:16) U H ( x − y ) f ( x − y ) − U H ( x ) f ( x ) (cid:17)(cid:12)(cid:12)(cid:12) p dx (cid:19) p = 0 . Here, D p := U H W p U is a matrix weight by Lemma 2.1.Proof. We use some ideas from [13, Proposition 3.6]. Assume that
F ⊂ L p ( W ) satisfies( a )–( c ). For any f ∈ L p ( W ), denote by ˜ f := U H f and ˜ F := { ˜ f } f ∈F . Then condition ( c ∗ )is equivalent to the equicontinuity of ˜ F ⊂ L p ( D ), that is,lim r → sup ˜ f ∈ ˜ F sup y ∈ B (0 ,r ) k τ y ˜ f − ˜ f k L p ( D ) = 0 . Moreover, from the orthogonality of U , it follows that (cid:12)(cid:12) D p ˜ f (cid:12)(cid:12) = (cid:12)(cid:12) U H W p U U H f (cid:12)(cid:12) = (cid:12)(cid:12) W p f (cid:12)(cid:12) . This via ( a ) and ( b ) for F shows that ˜ F ⊂ L p ( D ) is also bounded and uniformly vanishesat infinity. Observe that if ˜ F ⊂ L p ( D ) is totally bounded, so does F ⊂ L p ( W ) (indeed,the converse is also true). It suffices to verify the total boundedness of ˜ F ⊂ L p ( D ), thatis, to find a finite ǫ -net of ˜ F for each fixed ǫ > ≤ i ≤ d , we have | D p ˜ f | = | D ( λ p i )˜ f | ≥ λ p i | ˜ f i | , which impliesthat { ˜ f i } f ∈F ⊂ L p ( λ i ) satisfies conditions ( a )–( c ) of Theorem A with ω replaced by λ i ,where λ i ∈ L ( R n ) is a nonnegative eigenvalue function and ˜ f := ( ˜ f , · · · , ˜ f d ) T . Thenwe obtain that { ˜ f i } f ∈F ⊂ L p ( λ i ) is totally bounded by Theorem A.Hence, given ǫ >
0, for every f ∈ F and 1 ≤ i ≤ d , there exists g i ∈ L p ( λ i ) such that k ˜ f i − g i k L p ( λ i ) < ǫ . Let g := ( g , · · · , g d ) T . Then by the equivalence of norms in C d andour choice of the g i ’s, we have (cid:13)(cid:13) ˜ f − g (cid:13)(cid:13) L p ( D ) ≈ d X i =1 k ˜ f i − g i k L p ( λ i ) . ǫ, where implicit constants depend only on p and d . Finally, by the total boundedness of { ˜ f i } f ∈F , we conclude that { g } ⊂ L p ( D ) is a finite ǫ -net of ˜ F . This completes the proofof Theorem 3.6. (cid:3) Remark 3.7.
When d = 1 and W ( x ) = ω ( x ), we have D p ( x ) = ω p ( x ) and U ( x ) = 1.In this case, both Theorem 3.6 ( c ∗ ) and Corollary 3.4 ( c ) become Theorem A ( c ). Finally, we end this section with an application in degenerate Sobolev spaces withmatrix weights. Let W be an invertible matrix weight and set v := k W k op . For p ∈ [1 , ∞ ),the degenerate Sobolev space W ,pW ( R n ) due to Cruz-Uribe–Moen–Rodney [13] is definedas the set of all f ∈ W , ( R n ) such that k f k W ,pW ( R n ) := k f k L p ( v ) + k∇ f k L p ( W ) < ∞ , where ∇ f is the gradient of f . The degenerate Sobolev space W ,pW ( R n ) is an extensionof scalar weighted Sobolev spaces. From Theorem A and Corollary 3.4, we have thefollowing compactness criterion in W ,pW ( R n ), which is an extension of [22, Corollary 9]and [1, Theorem 12]. Corollary 3.8.
A subset
F ⊂ W ,pW ( R n ) is totally bounded if the following conditionshold: ( a ) F is bounded, i.e. sup f ∈F k f k W ,pW ( R n ) < ∞ ;( b ) F uniformly vanishes at infinity, that is, lim R →∞ sup f ∈F k f χ B c (0 ,R ) k W ,pW ( R n ) = 0;( c ) F is equicontinuous, that is, lim r → sup f ∈F sup y ∈ B (0 ,r ) k τ y f − f k W ,pW ( R n ) = 0 . Proof.
First, note that
F ⊂ W ,pW ( R n ) satisfies ( a )–( c ) if and only if F ⊂ L p ( v ) satisfies( a )–( c ) of Theorem A and ∇ ( F ) := {∇ f } f ∈F ⊂ L p ( W ) satisfies ( a )–( c ) of Corollary3.4. Then since W is an invertible matrix weight, by Theorem A and Corollary 3.4, weobtain that both F ⊂ L p ( v ) and ∇ ( F ) ⊂ L p ( W ) are totally bounded. Corollary 3.8 thenfollows from the fact that a set in metric spaces is totally bounded if and only if it isCauchy-precompact, that is, every sequence admits a Cauchy subsequence; see [41, p.262]. (cid:3) Compactness criteria on metric measure spaces
This section is devoted to the study of totally bounded sets in matrix weighted Lebesguespaces on metric measure spaces. We begin with some basic facts about metric measurespaces in [16, 20].Let (
X, d, µ ) be a metric measure space equipped with a metric d and a positive Borelregular measure µ . Let B ( x, r ) := { y ∈ X : d ( x, y ) < r } be the ball of the radius r > x ∈ X . We assume that the measure ofevery open nonempty set is strictly positive, and that the measure of every bounded setis finite. Definition 4.1.
A metric space is proper if every closed bounded set is compact.We remark that since every bounded set in a geometrically doubling metric space istotally bounded (see [23, Lemma 2.3]), a geometrically doubling metric space is properif and only if it is complete via the Hausdorff criterion.
ATRIX WEIGHTED KOLMOGOROV-RIESZ’S COMPACTNESS THEOREM 11
Definition 4.2.
Let (
X, d, µ ) be a metric measure space. The measure µ is said to becontinuous with respect to the metric d if for any x ∈ X and r > y → x µ [ B ( x, r )∆ B ( y, r )] = 0 , where A ∆ B stands for the symmetric difference of sets A and B . We call such a measuremetrically continuous for short, when no confusions can rise.From Definition 4.2, we have the following lemma. Lemma 4.3. If µ is metrically continuous, then for every compact set K ⊂ X and r > , inf x ∈ K µ [ B ( x, r )] > . Proof.
First, since µ is metrically continuous, then from Definition 4.2, we deduce thatfor any x, y ∈ X and r > (cid:12)(cid:12) µ [ B ( x, r )] − µ [ B ( y, r )] (cid:12)(cid:12) ≤ µ [ B ( x, r )∆ B ( y, r )] , which implies that for any given r >
0, the map x µ [ B ( x, r )] is continuous. Lemma4.3 then follows from the extreme value theorem. (cid:3) We now recall a vector-valued version of the classical Arzel´a-Ascoli theorem; see [17,Lemma 2.1] and [22, Theorem 2].
Lemma 4.4.
Let K be a compact topological space and C ( K, C d ) be the space of C d -valued continuous functions on K with the topology of uniform convergence. A subset F of C ( K, C d ) is totally bounded if and only if the following conditions are valid: ( a ) F is pointwise bounded, i.e. sup f ∈F | f ( x ) | < ∞ , ∀ x ∈ K ;( b ) F is equicontinuous, that is, for every x ∈ K and ǫ > , there is a neighborhood U of x such that | f ( x ) − f ( y ) | < ǫ, ∀ y ∈ U, f ∈ F . Now we give some necessary definitions and notations of matrix weights on metricmeasure spaces. A matrix function on X is a map W : X → M d . We say that it is µ -measurable if each component of W is a µ -measurable function on X , and invertible ifdet W ( x ) = 0 µ -a.e. and so W − exists.By a matrix weight on X we mean a µ -measurable matrix function W : X → S d suchthat k W k op ∈ L ( X, d, µ ). Equivalently, each eigenvalue function λ i ∈ L ( X, d, µ ) , ≤ i ≤ d . Let ρ := { ρ x } x ∈ X be a family of norms on C d , where for each x ∈ X , ρ x : C d → R + .Define the weighted Lebesgue space L p ( ρ, µ ) for p ∈ (0 , ∞ ) be the class of all µ -measurablevector-valued functions f : X → C d such that k f k pL p ( ρ,µ ) := Z X [ ρ x ( f ( x ))] p dµ ( x ) < ∞ , where we always assume that ρ x ( f ( x )) is a µ -measurable function on X for any µ -measurable vector-valued function f .Now we present our main result in this section as follows, in which we replace thetranslation operator by the average operator and apply Lemma 4.4, inspired by [20,Theorem 3.1]. Theorem 4.5.
Assume that ( X, d, µ ) is a proper metric measure space with a metricallycontinuous measure µ . Let < p < ∞ , ρ := { ρ x } x ∈ X be a family of norms on C d such that W x is an invertible matrix weight satisfying (2.3) for every x ∈ X and k W x k pop , k W − x k p ′ op ∈ L ( X, d, µ ) . A subset F of L p ( ρ, µ ) is totally bounded if the following conditions hold: ( a ) F is bounded, i.e. sup f ∈F k f k L p ( ρ,µ ) < ∞ ;( b ) F uniformly vanishes at infinity, that is, for some x ∈ X , lim R →∞ sup f ∈F k f χ X \ B ( x ,R ) k L p ( ρ,µ ) = 0;( c ) F is equicontinuous, that is, lim r → sup f ∈F k S r f − f k L p ( ρ,µ ) = 0 . Here, S r denotes the average operator: S r f ( x ) := 1 µ [ B ( x, r )] Z B ( x,r ) f ( y ) dµ ( y ) . Proof.
Assume that
F ⊂ L p ( ρ, µ ) satisfies ( a )–( c ). Given ǫ >
0, to prove the totalboundedness of F , it suffices to find a finite ǫ -net of F . By condition ( b ), there exists R > f ∈F k f − f χ B ( x ,R ) k L p ( ρ,µ ) < ǫ . (4.1)Moreover, by condition ( c ), there exists r ∈ (0 , R ) such thatsup f ∈F k S r f − f k L p ( ρ,µ ) < ǫ . (4.2)Then by (4.1) and (4.2), we havesup f ∈F k ( S r f ) χ B ( x ,R ) − f k L p ( ρ,µ ) ≤ sup f ∈F k ( S r f ) χ B ( x ,R ) − f χ B ( x ,R ) k L p ( ρ,µ ) + sup f ∈F k f − f χ B ( x ,R ) k L p ( ρ,µ ) < ǫ . So we only need to show that { ( S r f ) χ B ( x ,R ) } f ∈F has a finite ǫ -net.Next, we turn to verify that { S r f } f ∈F is pointwise bounded and equicontinuous on¯ B ( x , R ), where ¯ B ( x , R ) := { y ∈ X : d ( x , y ) ≤ R } is a closed bounded subset of X , and hence is compact by Definition 4.1. From Lemma2.2, we have the following estimate for any f ∈ L p ( ρ, µ ), Z B ( x , R ) | f ( y ) | dµ ( y )= Z B ( x , R ) | W − y W y f ( y ) | dµ ( y ) ≤ Z B ( x , R ) k W − y k op | W y f ( y ) | dµ ( y ) ≤ (cid:18) Z B ( x , R ) k W − y k p ′ op dµ ( y ) (cid:19) p ′ (cid:18) Z B ( x , R ) | W y f ( y ) | p dµ ( y ) (cid:19) p ≈ (cid:18) Z B ( x , R ) k W − y k p ′ op dµ ( y ) (cid:19) p ′ (cid:18) Z B ( x , R ) (cid:2) ρ y ( f ( y )) (cid:3) p dµ ( y ) (cid:19) p ATRIX WEIGHTED KOLMOGOROV-RIESZ’S COMPACTNESS THEOREM 13 . (cid:18) Z B ( x , R ) k W − y k p ′ op dµ ( y ) (cid:19) p ′ k f k L p ( ρ,µ ) . (4.3)It follows that for any f ∈ F and any fixed x ∈ ¯ B ( x , R ), | S r f ( x ) | . µ [ B ( x, r )] Z B ( x,r ) | f ( y ) | dµ ( y ) . µ [ B ( x, r )] Z B ( x , R ) | f ( y ) | dµ ( y ) . sup f ∈F k f k L p ( ρ,µ ) µ [ B ( x, r )] (cid:18) Z B ( x , R ) k W − y k p ′ op dµ ( y ) (cid:19) p ′ , (4.4)where we use the fact that B ( x, r ) ⊂ B ( x , R ) when x ∈ ¯ B ( x , R ). Since k W − y k p ′ op ∈ L ( X, d, µ ), then by condition ( a ), we obtain that { S r f } f ∈F is pointwise bounded on¯ B ( x , R ).Furthermore, for any f ∈ F and any fixed x ∈ ¯ B ( x , R ), by (4.4), we have the followingestimate that for any y ∈ ¯ B ( x , R ), | S r f ( y ) − S r f ( x ) |≤ (cid:12)(cid:12)(cid:12)(cid:12) µ [ B ( y, r )] Z B ( y,r ) f ( z ) dµ ( z ) − µ [ B ( x, r )] Z B ( y,r ) f ( z ) dµ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) µ [ B ( x, r )] Z B ( y,r ) f ( z ) dµ ( z ) − µ [ B ( x, r )] Z B ( x,r ) f ( z ) dµ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) . µ [ B ( x, r )∆ B ( y, r )] µ [ B ( x, r )] µ [ B ( y, r )] Z B ( y,r ) | f ( z ) | dµ ( z ) + 1 µ [ B ( x, r )] Z B ( x,r )∆ B ( y,r ) | f ( z ) | dµ ( z ) . µ [ B ( x, r )∆ B ( y, r )] µ [ B ( x, r )] µ [ B ( y, r )] Z B ( x , R ) | f ( z ) | dµ ( z ) + 1 µ [ B ( x, r )] Z B ( x,r )∆ B ( y,r ) | f ( z ) | dµ ( z ) . µ [ B ( x, r )∆ B ( y, r )] µ [ B ( x, r )] µ [ B ( y, r )] sup f ∈F k f k L p ( ρ,µ ) (cid:18) Z B ( x , R ) k W − z k p ′ op dµ ( z ) (cid:19) p ′ + 1 µ [ B ( x, r )] Z B ( x,r )∆ B ( y,r ) | f ( z ) | dµ ( z ) . (4.5)Since µ is metrically continuous, then by (4.3), (4.4) and Lemma 4.3, for any 0 < h ≤ r , x, y ∈ ¯ B ( x , R ) and f ∈ F , a direct calculation yields that Z B ( x,r )∆ B ( y,r ) | f ( z ) | dµ ( z ) ≤ Z B ( x,r )∆ B ( y,r ) | f ( z ) − S h f ( z ) | dµ ( z ) + Z B ( x,r )∆ B ( y,r ) | S h f ( z ) | dµ ( z ) ≤ Z B ( x , R ) | f ( z ) − S h f ( z ) | dµ ( z ) + Z B ( x,r )∆ B ( y,r ) | S h f ( z ) | dµ ( z ) . (cid:18) Z B ( x , R ) k W − z k p ′ op dµ ( z ) (cid:19) p ′ sup f ∈F k S h f − f k L p ( ρ,µ ) + µ [ B ( x, r )∆ B ( y, r )]inf z ∈ ¯ B ( x , R ) µ [ B ( z, h )] (cid:18) Z B ( x , R ) k W − z k p ′ op dµ ( z ) (cid:19) p ′ sup f ∈F k f k L p ( ρ,µ ) , (4.6)where we use the fact that B ( z, h ) ⊂ B ( x , R ) when z ∈ B ( x , R ), and implicit con-stants depend only on d . Then by the arbitrariness of h , condition ( c ), (4.5) and (4.6),we obtain that { S r f } f ∈F is equicontinuous on ¯ B ( x , R ).So from Lemma 4.4 we conclude that { S r f } f ∈F is totally bounded in C ( ¯ B ( x , r ) , C d ).It follows that there exists { f k } Nk =1 ⊂ F such that { S r f k } Nk =1 is an ǫA -net of { S r f } f ∈F forgiven ǫ , where A := 3 (cid:0) R B ( x ,R ) k W x k pop dµ ( x ) (cid:1) p .Hereafter, we shall show that { ( S r f k ) χ B ( x ,R ) } Nk =1 is a finite ǫ -net of { ( S r f ) χ B ( x ,R ) } f ∈F in L p ( ρ, µ ). Note that by Lemma 2.2, k ( S r f ) χ B ( x ,R ) − ( S r f k ) χ B ( x ,R ) k L p ( ρ,µ ) ≤ (cid:18) Z B ( x ,R ) | W x ( S r f ( x ) − S r f k ( x )) | p dµ ( x ) (cid:19) p ≤ (cid:18) Z B ( x ,R ) k W x k pop | S r f ( x ) − S r f k ( x ) | p dµ ( x ) (cid:19) p ≤ (cid:18) Z B ( x ,R ) k W x k pop dµ ( x ) (cid:19) p sup x ∈ ¯ B ( x ,R ) | S r f ( x ) − S r f k ( x ) | < ǫ , which finishes the proof of Theorem 4.5. (cid:3) Remark 4.6. (i) By using the same argument in Theorem 4.5 with some minor changes,one can prove that Theorem 4.5 also holds for p = 1 under the additional assumptionsthat W x is an invertible matrix weight satisfying (2.3) for every x ∈ X and k W − x k op ∈ L ∞ loc ( X, d, µ ) . (ii) The assumption on ρ x in Theorem 4.5 is necessary for our method. Since ourmethod relies on the structure of Banach function spaces (see [20, Definition 2.1]), al-though matrix weighted Lebesgue spaces are not Banach lattices (see [6]). Moreover,Tsuji’s method is invalid here to relax the range of the exponent p ∈ (0 , ∞ ).As an application, we obtain the following compactness criterion in matrix weightedLebesgue spaces on ( R n , | · | , µ ) with a metrically continuous measure µ , by applyingTheorem 4.5 with ρ x ( · ) := | W p ( x ) · | . This extends the corresponding results of [9,Theorem 5], [43, Lemma 4.1] and [7, Proposition 2.9]. Corollary 4.7.
Let ≤ p < ∞ , ( R n , | · | , µ ) be the Euclidean metric measure spacewith a metrically continuous measure µ . Assume that W is an invertible matrix weightsatisfying ( i ) k W − k op ∈ L ∞ loc ( R n , | · | , µ ) when p = 1 ; ( ii ) k W − k p ′ p op ∈ L ( R n , | · | , µ ) when p ∈ (1 , ∞ ) .Define k f k pL p ( W,µ ) := Z R n (cid:12)(cid:12) W p ( x ) f ( x ) (cid:12)(cid:12) p dµ ( x ) . A subset F of L p ( W, µ ) is totally bounded if the following conditions are valid: ( a ) F is bounded, i.e. sup f ∈F k f k L p ( W,µ ) < ∞ ; ATRIX WEIGHTED KOLMOGOROV-RIESZ’S COMPACTNESS THEOREM 15 ( b ) F uniformly vanishes at infinity, that is, lim R →∞ sup f ∈F k f χ B c (0 ,R ) k L p ( W,µ ) = 0;( c ) F is equicontinuous, that is, lim r → sup f ∈F k S r f − f k L p ( W,µ ) = 0 . Clearly, by Definition 2.3, if W is a matrix A p weight for p ∈ (1 , ∞ ), then W satisfiesthe assumption in Corollary 4.7 on ( R n , | · | , m ).5. A characterization of compactness on R n In this section, we give a necessary and sufficient condition for total boundedness ofsubsets in matrix weighted Lebesgue spaces. Before that, we need some lemmas.For any 0 < r < ∞ , let S r be the average operator on R n defined by S r f ( x ) := 1 | B ( x, r ) | Z B ( x,r ) f ( y ) dy, ∀ f ∈ L p ( W ) . Then we have the following useful lemma.
Lemma 5.1.
Let < p < ∞ . If W ∈ A p , then k S r f k L p ( W ) . k f k L p ( W ) , ∀ f ∈ L p ( W ) , where the implicit constant depends only on p and d .Proof. Note that for any f ∈ L p ( W ) and 0 < r < ∞ , (cid:12)(cid:12)(cid:12)(cid:12) W p ( x ) | B ( x, r ) | Z B ( x,r ) f ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) . | B ( x, r ) | Z B ( x,r ) (cid:12)(cid:12) W p ( x ) f ( y ) (cid:12)(cid:12) dy ≈ | B ( x, r ) | Z B ( x,r ) (cid:12)(cid:12) W p ( x ) W − p ( y ) W p ( y ) f ( y ) (cid:12)(cid:12) dy . M w (cid:0) W p f (cid:1) ( x ) . (5.1)Then from (5.1) and Lemma 2.5, it follows that k S r f k L p ( W ) . k M ω (cid:0) W p f (cid:1) k L p ( R n ) . k W p f k L p ( R n , C d ) ≈ k f k L p ( W ) . This completes the proof of Lemma 5.1. (cid:3)
The following is a vector-valued extension of the Lebesgue differentiation theorem inthe setting of matrix weights.
Lemma 5.2.
Let < p < ∞ . If W ∈ A p , then for any f ∈ L p ( W ) , lim r → | S r f ( x ) − f ( x ) | = 0 a.e. x ∈ R n . Proof.
First, for any f := ( f , · · · , f d ) T ∈ L p ( W ), it suffices to show that f i ∈ L ( R n )for each 1 ≤ i ≤ d . Since W ∈ A p , then by (2.2), | f | p = | W − p W p f | p ≤ k W − p k pop | W p f | p = k W − k op | W p f | p . It follows that | f | p k W − k − op ≤ | W p f | p . From Lemma 2.4, we conclude that k W − k − op is a scalar A p weight, and hence | f | ∈ L ( R n ), which implies that f i ∈ L ( R n ) for each 1 ≤ i ≤ d .Hence, by the classical Lebesgue differentiation theorem, for each 1 ≤ i ≤ d , we havelim r → | S r f i ( x ) − f i ( x ) | = 0 a.e. x ∈ R n . Lemma 5.2 then follows from the fact that for any x ∈ R n , | S r f ( x ) − f ( x ) | ≤ d max ≤ i ≤ d | S r f i ( x ) − f i ( x ) | . (cid:3) We now present a characterization for total boundedness of subsets in L p ( W ) when W ∈ A p . Theorem 5.3.
Let < p < ∞ , and W ∈ A p . A subset F of L p ( W ) is totally bounded ifand only if the following conditions hold: ( a ) F is bounded, i.e. sup f ∈F k f k L p ( W ) < ∞ ;( b ) F uniformly vanishes at infinity, that is, lim R →∞ sup f ∈F k f χ B c (0 ,R ) k L p ( W ) = 0;( c ) F is equicontinuous, that is, lim r → sup f ∈F k S r f − f k L p ( W ) = 0 . Proof.
The sufficiency is due to Corollary 4.7. We now proof the necessity. Assume that
F ⊂ L p ( W ) is totally bounded. Then for any given ǫ >
0, there exists { f k } Nk =1 ⊂ F suchthat { f k } Nk =1 is an ǫ -net of F , which implies that for any f ∈ F , there exists f k such that k f − f k k L p ( W ) < ǫ. Clearly, ( a ) is true. As for ( b ), for each 1 ≤ k ≤ N , since f k ∈ L p ( W ), by themonotone convergence theorem, there exists R k > k f k χ B c (0 ,R k ) k L p ( W ) < ǫ. Set R := max { R k : 1 ≤ k ≤ N } . It follows that for given f ∈ F , k f χ B c (0 ,R ) k L p ( W ) ≤ k f − f k k L p ( W ) + k f k χ B c (0 ,R ) k L p ( W ) < ǫ, which implies ( b ).As for ( c ), for each 1 ≤ k ≤ N , by W ∈ A p , Lemma 2.5 and (5.1), (cid:12)(cid:12) W p ( x ) (cid:0) S r f k ( x ) − f k ( x ) (cid:1)(cid:12)(cid:12) ≤ (cid:12)(cid:12) W p ( x ) S r f k ( x ) (cid:12)(cid:12) + (cid:12)(cid:12) W p ( x ) f k ( x ) (cid:12)(cid:12) ≤ M ω (cid:0) W p f k (cid:1) ( x ) + (cid:12)(cid:12) W p ( x ) f k ( x ) (cid:12)(cid:12) ∈ L p ( R n ) . Then by Lemma 5.2 and Lebesgue’s dominated convergence theorem, there exists r > h ≤ r , max ≤ k ≤ N k S h f k − f k k L p ( W ) < ǫ. From Lemma 5.1, it follows that k S h f − f k L p ( W ) ≤ k S h f − S h f k k L p ( W ) + k S h f k − f k k L p ( W ) + k f k − f k L p ( W ) . k f k − f k L p ( W ) + k S h f k − f k k L p ( W ) . ǫ, where implicit constants depend only on p and d . This implies ( c ) and completes theproof of Theorem 5.3. (cid:3) ATRIX WEIGHTED KOLMOGOROV-RIESZ’S COMPACTNESS THEOREM 17
Acknowledgement
Yang is supported by the National Natural Science Foundation of China (Grant Nos.11971402 and 11871254). Zhuo is supported by the National Natural Science Foundationof China (Grant Nos. 1187110 and 11701174).
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Email address : [email protected] Dongyong Yang, School of Mathematical Sciences, Xiamen University, Xiamen 361005,China
Email address : [email protected] Ciqiang Zhuo(Corresponding author), School of Mathematics and Statistics, HunanNormal University, Changsha, Hunan 410081, China
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