Greedy bases in variable Lebesgue spaces
aa r X i v : . [ m a t h . F A ] O c t GREEDY BASES IN VARIABLE LEBESGUE SPACES
DAVID CRUZ-URIBE, SFO, EUGENIO HERN ´ANDEZ, AND JOS´E MAR´IA MARTELL
Abstract.
We compute the right and left democracy functions of admissiblewavelet bases in variable Lebesgue spaces defined on R n . As an application wegive Lebesgue type inequalities for these wavelet bases. We also show that ourtechniques can be easily modified to prove analogous results for weighted variableLebesgue spaces and variable exponent Triebel-Lizorkin spaces. Introduction
Let X be an infinite dimensional Banach space with norm k·k X and let B = { b j } ∞ j =1 be a Schauder basis for X : that is, if x ∈ X , then there exists a unique sequence { λ j } such that(1.1) x = ∞ X j =1 λ j b j . For each N = 1 , , , . . . we define the best N -term approximation of x ∈ X in termsof B as follows: σ N ( x ) = σ N ( x ; B ) := inf y ∈ Σ N k x − y k X where Σ N the set of all y ∈ X with at most N non-zero coefficients in their basisrepresentation.An important question in approximation theory is the construction of efficient algo-rithms for N -term approximation. One algorithm that has been extensively studied inrecent years is the so called greedy algorithm. Given x ∈ X and the coefficients (1.1),reorder the basis elements so that Date : October 1, 2014.2010
Mathematics Subject Classification.
Key words and phrases.
Greedy algorithm, non-linear approximation, wavelets, variable Lebesguespaces, Lebesgue type estimates.The first author is supported by the Stewart-Dorwart faculty development fund at Trinity Collegeand NSF grant 1362425. The second and third authors are supported in part by MINECO GrantMTM2010-16518 (Spain). The third author has been also supported by ICMAT Severo Ochoaproject SEV-2011-0087 (Spain) and he also acknowledges that the research leading to these resultshas received funding from the European Research Council under the European Union’s SeventhFramework Programme (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT. k λ j b j k X ≥ k λ j b j k X ≥ k λ j b j k X . . . (handling ties arbitrarily); we then define an N -term approximation by the (non-linear) operator G N : X → Σ N , G N ( x ) = N X k =1 λ j k b j k . Clearly, σ N ( x ) ≤ k x − G N ( x ) k X . We say that a basis is greedy if the oppositeinequality holds up to a constant: there exists C > x ∈ X and N > k x − G N ( x ) k X ≤ Cσ N ( x ) . Konyagin and Temlyakov [27] characterized greedy bases as those which are un-conditional and democratic. A basis is democratic if given any two index sets Γ , Γ ′ ,card(Γ) = card(Γ ′ ), (cid:13)(cid:13)(cid:13)(cid:13) X j ∈ Γ b j k b j k X (cid:13)(cid:13)(cid:13)(cid:13) X ≈ (cid:13)(cid:13)(cid:13)(cid:13) X j ∈ Γ ′ b j k b j k X (cid:13)(cid:13)(cid:13)(cid:13) X . Wavelet systems form greedy bases in many function and distribution spaces. Forexample, Temlyakov [33] proved that any wavelet basis p -equivalent to the Haar basisis a greedy basis in L p ( R n ). However, wavelet bases are not greedy in other functionspaces. For example, it was shown in [18] that if Φ is a Young function such that theOrlicz space L Φ is not L p , 1 < p < ∞ , then wavelet bases are not greedy because theyare not democratic. (Earlier, Soardi [32] proved that wavelet bases are unconditionalin L Φ .)In this paper our goal is to extend this result to the variable Lebesgue spacesand other related function spaces. Intuitively, given an exponent function p ( · ), thevariable Lebesgue space L p ( · ) ( R n ) consists of all functions f such that Z R n | f ( x ) | p ( x ) dx < ∞ . (See Section 2 below for a precise definition.) These spaces are a generalization ofthe classical L p spaces, and have applications in the study of PDEs and variationalintegrals with non-standard growth conditions. For the history and properties ofthese spaces we refer to [3, 9].Many wavelet bases form unconditional bases on the variable Lebesgue spaces: seeTheorem 2.1 below. However, unless p ( · ) = p is constant, they are never democraticin L p ( · ) ( R n ). When n = 1 this was proved by Kopaliani [28] and for general n itfollows from Theorem 1.1 below. In this paper we quantify the failure of democracyby computing precisely the right and left democracy functions of admissible wavelet REEDY BASES IN VARIABLE LEBESGUE SPACES 3 bases in L p ( · ) ( R n ). For a basis B = { b j } ∞ j =1 in a Banach space X , we define the rightand left democracy functions of X (see also [11, 23]) as: h r ( N ) = sup card(Γ)= N (cid:13)(cid:13)(cid:13)(cid:13) X j ∈ Γ b j k b j k X (cid:13)(cid:13)(cid:13)(cid:13) X , h l ( N ) = inf card(Γ)= N (cid:13)(cid:13)(cid:13)(cid:13) X j ∈ Γ b j k b j k X (cid:13)(cid:13)(cid:13)(cid:13) X To state our main result, given a variable exponent p ( · ) define p + = ess sup x ∈ R n p ( x )and p − = ess inf x ∈ R n p ( x ) . Theorem 1.1.
Given an exponent function p ( · ) , suppose < p − ≤ p + < ∞ and theHardy-Littlewood maximal operator is bounded on L p ( · ) ( R n ) . Let Ψ be an admissibleorthonormal wavelet family (see below for the precise definition). The right and leftdemocracy functions of Ψ in L p ( · ) ( R n ) satisfy h r ( N ) ≈ N /p − , h l ( N ) ≈ N /p + , N = 1 , , , . . . As an immediate corollary to Theorem 1.1 we get a Lebesgue-type estimate forwavelet bases on variable Lebesgue spaces. The proof follows from Wojtaszczyk [34,Theorem 4].
Corollary 1.2.
With the hypotheses of Theorem 1.1, we have that for all f ∈ L p ( · ) ( R n ) , k f − G N ( f ) k L p ( · ) ( R n ) ≤ CN p − − p + σ N ( f, Ψ) , and this estimate is the best possible. Our arguments readily adapt to prove analogs of Theorem 1.1 (and so also of Corol-lary 1.2) for other variable exponent spaces, in particular for the weighted variableLebesgue spaces [4] and for variable exponent Triebel-Lizorkin spaces [10, 25, 26, 36].The statement of these results require a number of preliminary definitions, and so wedefer these until after the proof of Theorem 1.1.The remainder of this paper is organized as follows. In Section 2 we give a numberof preliminary results regarding the variable Lebesgue spaces that are needed in ourmain proof. In Section 3 we prove Theorem 1.1. In Sections 4 and 5 we state and provethe corresponding result for weighted variable Lebesgue spaces and variable exponentTriebel-Lizorkin spaces. Throughout this paper, our notation will be standard ordefined as needed. If we write A . B , we mean that A ≤ CB , where the constant C depends only on the dimension n and the underlying exponent function p ( · ). If A . B and B . A , we write A ≈ B . The letter C will denote a constant that maychange at each appearance. DAVID CRUZ-URIBE, SFO, EUGENIO HERN ´ANDEZ, AND JOS´E MAR´IA MARTELL Variable Lebesgue spaces
Basic properties.
We begin with some basic definitions and results about variableLebesgue spaces. For proofs and further information, see [3, 9, 13, 29].Let P = P ( R n ) be the collection of exponent functions: that is, all measurablefunctions p ( · ) : R n → [1 , ∞ ). We define the variable Lebesgue space L p ( · ) = L p ( · ) ( R n )to be the family of all measurable functions f such that for some λ > Z R n (cid:18) | f ( x ) | λ (cid:19) p ( x ) dx < ∞ . This becomes a Banach function space with respect to the Luxemburg norm k f k p ( · ) = inf ( λ > Z R n (cid:18) | f ( x ) | λ (cid:19) p ( x ) dx ≤ ) . When p ( · ) = p is constant, then L p ( · ) = L p with equality of norms.To measure the oscillation of p ( · ), given any set E ⊂ R n , we define p + ( E ) = ess sup x ∈ E p ( x ) , p − ( E ) = ess inf x ∈ E p ( x ) . For brevity we write p + = p + ( R n ), p − = p − ( R n ).When p + < ∞ , we have the following useful integral estimate: k f k p ( · ) is the uniquevalue such that(2.1) Z R n (cid:18) | f ( x ) |k f k p ( · ) (cid:19) p ( x ) dx = 1 . Given an exponent p ( · ), 1 < p − ≤ p + < ∞ we define the conjugate exponent p ′ ( · )pointwise by 1 p ( x ) + 1 p ′ ( x ) = 1 . Then functions f ∈ L p ( · ) ( R n ) and g ∈ L p ′ ( · ) ( R n ) satisfy H¨older’s inequality: Z R n | f ( x ) g ( x ) | dx ≤ k f k p ( · ) k g k p ′ ( · ) ;moreover, L p ′ ( · ) ( R n ) is the dual space of L p ( · ) ( R n ) and k f k p ( · ) ≈ sup k g k p ′ ( · ) =1 (cid:12)(cid:12)(cid:12) Z R n f ( x ) g ( x ) dx (cid:12)(cid:12)(cid:12) . REEDY BASES IN VARIABLE LEBESGUE SPACES 5
The Hardy-Littlewood maximal operator.
To do harmonic analysis on variableLebesgue spaces, it is necessary to assume some regularity on the exponent p ( · ). Oneapproach (taken from [3]) is to express this regularity in terms of the boundednessof the Hardy-Littlewood maximal operator. Given a locally integrable function f ,define M f by M f ( x ) = sup Q ∋ x − Z Q | f ( y ) | dy, where the supremum is taken over all cubes Q with sides parallel to the coordinateaxes. If the maximal operator is bounded on L p ( · ) we will write p ( · ) ∈ M P .The following are basic properties of the maximal operator on variable Lebesguespaces. For complete information, see [3, 9]. By Chebyschev’s inequality, if M isbounded, then it also satisfies the weak-type inequality(2.2) k tχ { x : Mf ( x ) >t } k p ( · ) ≤ C k f k p ( · ) , t > . A necessary condition for p ( · ) ∈ M P is that p − >
1. An important sufficient conditionis that p ( · ) is log-H¨older continuous locally: there exists C > | p ( x ) − p ( y ) | ≤ C − log( | x − y | ) , | x − y | < / p ∞ and C ∞ > | p ( x ) − p ∞ | ≤ C ∞ log( e + | x | ) . These conditions are not necessary for the maximal operator to be bounded on L p ( · ) ,but they are sharp in the sense that they are best possible pointwise continuityconditions guaranteeing that M is bounded on L p ( · ) . Weighted norm inequalities.
There is a close connection between the variableLebesgue spaces and the theory of weighted norm inequalities. Here we give somebasic information on weights; for more information, see [3, 12, 16].By a weight we mean a non-negative, locally integrable function. For 1 < p < ∞ ,we say that a weight w is in the Muckenhoupt class A p if[ w ] A p = sup Q (cid:18) − Z Q w ( x ) dx (cid:19) (cid:18) − Z Q w ( x ) − p ′ dx (cid:19) p − < ∞ , where − R Q g ( x ) dx = | Q | − R Q g ( x ) dx. When p = 1, we say that w ∈ A if[ w ] A = (cid:18) − Z Q w ( y ) dy (cid:19) ess sup x ∈ Q w ( x ) − < ∞ . DAVID CRUZ-URIBE, SFO, EUGENIO HERN ´ANDEZ, AND JOS´E MAR´IA MARTELL
Equivalently, w ∈ A if M w ( x ) ≤ [ w ] A w ( x ) almost everywhere, where M is theHardy-Littlewood maximal operator. Define A ∞ = S p ≥ A p . If w ∈ A ∞ , then thereexist constants C, δ > Q and E ⊂ Q , w ( E ) w ( Q ) ≤ C (cid:18) | E || Q | (cid:19) δ , where w ( E ) = R E w ( x ) dx . Wavelets.
To state our results precisely we need a few definitions on wavelets; forcomplete information we refer the reader to [20]. Given the collection of dyadic cubes D = { Q j,k = 2 − j ([0 , n + k ) : j ∈ Z , k ∈ Z n } , the functions Ψ = { ψ , . . . , ψ L } ⊂ L ( R n ) form an orthonormal wavelet family if { ψ lQ } = (cid:8) ψ lQ j,k ( x ) = 2 j n/ ψ l (2 j x − k ) : j ∈ Z , k ∈ Z n , ≤ l ≤ L (cid:9) is an orthonormal basis of L ( R n ).Define the square function W Ψ f = (cid:18) L X l =1 X Q ∈D |h f, ψ lQ i| | Q | − χ Q (cid:19) / . We will say that a wavelet family Ψ is admissible if for 1 < p < ∞ and every w ∈ A p , kW Ψ f k L p ( w ) ≈ k f k L p ( w ) . Admissible wavelets on the real line include the Haar system [24], spline wavelets[14], the compactly supported wavelets of Daubechies [8], Lemari´e-Meyer wavelets[30, 35], and smooth wavelets in the class R [15, 20].An important consequence of the boundedness of the maximal operator on L p ( · ) isthat in this case wavelets form an unconditional basis. Theorem 2.1.
Given p ( · ) , suppose < p − ≤ p + < ∞ and p ( · ) ∈ M P . If Ψ is anadmissible orthonormal wavelet family, then it is an unconditional basis for L p ( · ) ( R n ) and kW Ψ f k p ( · ) ≈ k f k p ( · ) . Theorem 2.1 was proved in [5, Theorem 4.27] using the theory of Rubio de Franciaextrapolation. The result is stated with the stronger hypothesis that p ( · ) is log-H¨older continuous, but the extrapolation argument given there works with the weakerassumptions used here (see [3, Corollary 5.32]). This result was also proved byIzuki [22] and by Kopaliani [28] on the real line. REEDY BASES IN VARIABLE LEBESGUE SPACES 7 Proof of Theorem 1.1
In this section we give the proof of Theorem 1.1. In order to avoid repeating detailsin the subsequent sections, we have written the proof in terms of a series of lemmasand propositions; this will allow us to prove our other results by indicating wherethis proof must be modified.
Lemma 3.1.
Given an exponent function p ( · ) ∈ P ( R n ) such that < p − ≤ p + < ∞ ,then for every cube Q , | Q | pQ ≤ k χ Q k p ( · ) , where p Q = − Z Q p ( x ) dx. When p ( · ) ∈ M P , this inequality is actually an equivalence: see [9]. For ourpurposes we only need this weaker result and so we include the short proof. Proof.
Fix a cube Q . If we define 1 p ′ Q = − Z Q p ′ ( x ) dx, then 1 /p Q + 1 /p ′ Q = 1. By Jensen’s inequality, (cid:18) | Q | (cid:19) p ′ Q = exp (cid:18) − Z Q log (cid:20) (cid:18) | Q | (cid:19) p ′ ( · ) (cid:21) dx (cid:19) ≤ − Z Q (cid:18) | Q | (cid:19) p ′ ( · ) dx. But then by H¨older’s inequality in the scale of variable Lebesgue spaces, | Q | pQ = | Q | (cid:18) | Q | (cid:19) p ′ Q ≤ | Q |− Z Q (cid:18) | Q | (cid:19) p ′ ( · ) dx ≤ k χ Q k p ( · ) k| Q | − /p ′ ( · ) χ Q k p ′ ( · ) . To complete the proof, note that Z Q (cid:0) | Q | − /p ′ ( x ) (cid:1) p ′ ( x ) dx = 1 , and so by (2.1), k| Q | − /p ′ ( · ) χ Q k p ′ ( · ) = 1. (cid:3) Lemma 3.2.
Given p ( · ) , suppose p ( · ) ∈ M P . Then for any cube Q and any set E ⊂ Q , | E || Q | ≤ M k χ E k p ( · ) k χ Q k p ( · ) , where M is the norm of the Hardy-Littlewood operator M on L p ( · ) ( R n ) . DAVID CRUZ-URIBE, SFO, EUGENIO HERN ´ANDEZ, AND JOS´E MAR´IA MARTELL
Proof.
Fix Q and E ⊂ Q . Then for every x ∈ Q , M ( χ E )( x ) ≥ | E || Q | . Since M is bounded on L p ( · ) ( R n ), by the weak-type inequality with t = | E || Q | (noticethat the constant C in the right hand side of (2.2) can be taken to be M ), | E || Q | k χ Q k p ( · ) ≤ M k χ E k p ( · ) . (cid:3) Lemma 3.3.
Given p ( · ) , suppose p ( · ) ∈ M P and < p − ≤ p + < ∞ . Then thereexist constants C, δ > such that given any cube Q and any set E ⊂ Q , k χ E k p ( · ) k χ Q k p ( · ) ≤ C (cid:18) | E || Q | (cid:19) δ . Proof.
Since p ( · ) ∈ M P and 1 < p − ≤ p + < ∞ , we have that p ′ ( · ) ∈ M P [3,Corollary 4.64]. Therefore, we can define a Rubio de Francia iteration algorithm [5,Section 2.1]: Rg ( x ) = ∞ X k =0 M k g ( x )2 k k M k kp ′ ( · ) , where k M k p ′ ( · ) is the operator norm of the maximal operator on L p ′ ( · ) and M g = | g | .Then g and Rg are comparable in size: | g ( x ) | ≤ Rg ( x ) and k Rg k p ′ ( · ) ≤ k g k p ′ ( · ) .Moreover, Rg ∈ A and [ Rg ] A ≤ k M k p ′ ( · ) . Therefore, there exist C, δ > Q and E ⊂ Q , Rg ( E ) Rg ( Q ) ≤ C (cid:18) | E || Q | (cid:19) δ . Now by duality and H¨older’s inequality, there exists g ∈ L p ′ ( · ) , k g k p ′ ( · ) = 1, suchthat k χ E k p ( · ) ≤ C Z R n χ E ( x ) g ( x ) dx ≤ CRg ( E ) ≤ C (cid:18) | E || Q | (cid:19) δ Rg ( Q ) ≤ C (cid:18) | E || Q | (cid:19) δ k χ Q k p ( · ) k Rg k p ′ ( · ) ≤ C (cid:18) | E || Q | (cid:19) δ k χ Q k p ( · ) . (cid:3) REEDY BASES IN VARIABLE LEBESGUE SPACES 9
We can now prove Theorem 1.1. We first make some reductions, and then dividethe proof into three propositions. First, we will do the proof for a single admissiblewavelet ψ, since considering a family of L wavelets will only introduce an additionalfinite sum and make the constants depend on L .Second, to prove Theorem 1.1 we need to estimate expressions of the form (cid:13)(cid:13)(cid:13)(cid:13) X Q ∈ Γ ψ Q k ψ Q k p ( · ) (cid:13)(cid:13)(cid:13)(cid:13) p ( · ) . for any finite set Γ of dyadic cubes. By Theorem 2.1 we have k ψ Q k p ( · ) ≈ k| Q | − / χ Q k p ( · ) = | Q | − / k χ Q k p ( · ) . Thus, again by Theorem 2.1,(3.1) (cid:13)(cid:13)(cid:13)(cid:13) X Q ∈ Γ ψ Q k ψ Q k p ( · ) (cid:13)(cid:13)(cid:13)(cid:13) p ( · ) ≈ (cid:13)(cid:13)(cid:13)(cid:13) X Q ∈ Γ ψ Q | Q | − / k χ Q k p ( · ) (cid:13)(cid:13)(cid:13)(cid:13) p ( · ) ≈ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) X Q ∈ Γ χ Q k χ Q k p ( · ) (cid:19) / (cid:13)(cid:13)(cid:13)(cid:13) p ( · ) . Therefore, it will be enough to show that the righthand expression satisfies the desiredinequalities. It is illuminating at this point to consider the special case where thecubes in Γ are pairwise disjoint. With this as a model we will then obtain the desiredestimate in the general case.
Proposition 3.4.
Given an exponent function p ( · ) , suppose < p − ≤ p + < ∞ and the Hardy-Littlewood maximal operator is bounded on L p ( · ) ( R n ) . Then thereexist constants such that given any collection Γ of pairwise disjoint dyadic cubes, card(Γ) = N , N /p + . (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) X Q ∈ Γ χ Q k χ Q k p ( · ) (cid:19) / (cid:13)(cid:13)(cid:13)(cid:13) p ( · ) . N /p − . Proof.
We will prove the first inequality; the second is proved in essentially the sameway, replacing p + by p − and reversing the inequalities. Fix a collection Γ withCard (Γ) = N . Since the cubes in Γ are disjoint, we have that (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) X Q ∈ Γ χ Q k χ Q k p ( · ) (cid:19) / (cid:13)(cid:13)(cid:13)(cid:13) p ( · ) = (cid:13)(cid:13)(cid:13)(cid:13) X Q ∈ Γ χ Q k χ Q k p ( · ) (cid:13)(cid:13)(cid:13)(cid:13) p ( · ) . We now estimate as follows: Z R n N − /p + X Q ∈ Γ χ Q k χ Q k p ( · ) ! p ( x ) dx = X Q ∈ Γ Z Q N − p ( x ) /p + k χ Q k − p ( x ) p ( · ) dx ≥ N − X Q ∈ Γ Z Q k χ Q k − p ( x ) p ( · ) dx = 1;the last inequality follows from (2.1). Therefore, by the definition of the L p ( · ) norm, N /p + ≤ (cid:13)(cid:13)(cid:13)(cid:13) X Q ∈ Γ χ Q k χ Q k p ( · ) (cid:13)(cid:13)(cid:13)(cid:13) p ( · ) . (cid:3) In general, the cubes in the collection Γ will not be disjoint. To overcome this, wewill show that we can linearize the square function S Γ ( x ) := (cid:18) X Q ∈ Γ χ Q k χ Q k p ( · ) (cid:19) / . Such linearization arguments were previously considered in [2, 17, 21]. Here, we willuse the technique of “lighted” and “shaded” cubes introduced in [18].
Proposition 3.5.
Given an exponent function p ( · ) , suppose < p − ≤ p + < ∞ and the Hardy-Littlewood maximal operator is bounded on L p ( · ) ( R n ) . Let Γ be anyfinite collection of dyadic cubes. Then there exists a sub-collection Γ min ⊂ Γ and acollection of pairwise disjoint sets { Light( Q ) } Q ∈ Γ min , such that Light( Q ) ⊂ Q and S Γ ( x ) ≈ X Q ∈ Γ min χ Light( Q ) ( x ) k χ Q k p ( · ) . In these inequalities the constants are independent of the set Γ .Proof. Fix a finite collection Γ and let Ω Γ = S Q ∈ Γ Q . For each x ∈ Ω Γ , let Q x ∈ Γbe the unique smallest cube that contains x . We immediately have that for every x ∈ Ω Γ , S Γ ( x ) ≥ χ Q x ( x ) k χ Q x k p ( · ) . We claim that the reverse inequality holds up to a constant. Indeed, let Q x = Q ⊂ Q ⊂ Q ⊂ Q ⊂ · · · be the sequence of all dyadic cubes that contain Q x . Then | Q j | = 2 jn | Q | , and byLemma 3.3, k χ Q k p ( · ) k χ Q j k p ( · ) ≤ C (cid:18) | Q || Q j | (cid:19) δ ≤ C − jnδ . REEDY BASES IN VARIABLE LEBESGUE SPACES 11
Hence, S Γ ( x ) ≤ ∞ X j =0 k χ Q j k p ( · ) ≤ C k χ Q x k p ( · ) ∞ X j =0 − jnδ = C χ Q x ( x ) k χ Q x k p ( · ) . This gives us the pointwise equivalence(3.2) S Γ ( x ) ≈ χ Q x ( x ) k χ Q x k p ( · ) . Let Γ min = { Q x : x ∈ Ω Γ } ; note that the cubes in Γ min may still not be pairwisedisjoint. To obtain a disjoint family we argue as in [18]. Given Q ∈ Γ, let Shade( Q ) = S { R : R ∈ Γ , R Q } and Light( Q ) = Q \ Shade ( Q ) . Then (see [18, Section 4.2.2])we have that Q ∈ Γ min if and only if Light( Q ) = ∅ , x ∈ Light( Q x ), the sets Light( Q )are pairwise disjoint, and [ Q ∈ Γ Q = [ Q ∈ Γ min Light( Q ) . If we combine this analysis with (3.2) we get(3.3) S Γ ( x ) ≈ X Q ∈ Γ min χ Light( Q ) ( x ) k χ Q k p ( · ) , where in the righthand sum there is at most one non-zero term for any x ∈ Ω Γ . (cid:3) We can now estimate the square function S Γ for an arbitrary finite set of dyadiccubes Γ. Proposition 3.6.
Given an exponent function p ( · ) , suppose < p − ≤ p + < ∞ andthe Hardy-Littlewood maximal operator is bounded on L p ( · ) ( R n ) . If Γ is a finite set ofdyadic cubes, card(Γ) = N , then N /p + . k S Γ k p ( · ) . N /p − . Proof.
By Proposition 3.5, to prove the righthand inequality it suffices to show that (cid:13)(cid:13)(cid:13)(cid:13) X Q ∈ Γ min χ Light( Q ) k χ Q k p ( · ) (cid:13)(cid:13)(cid:13)(cid:13) p ( · ) ≤ N /p − . By the definition of the L p ( · ) norm, this follows from the fact that Z R n N − /p − X Q ∈ Γ min χ Light( Q ) ( x ) k χ Q k p ( · ) ! p ( x ) dx = X Q ∈ Γ min Z Light( Q ) N − p ( x ) /p − k χ Q k − p ( x ) p ( · ) dx ≤ N X Q ∈ Γ min Z Q k χ Q k − p ( x ) p ( · ) dx = 1 N card(Γ min ) ≤ , where we have used that the sets Light( Q ) are disjoint, p ( x ) ≥ p − and (2.1).We now prove the lefthand inequality; again by Proposition 3.5 it suffices to showthat (cid:13)(cid:13)(cid:13)(cid:13) X Q ∈ Γ min χ Light( Q ) k χ Q k p ( · ) (cid:13)(cid:13)(cid:13)(cid:13) p ( · ) ≥ CN /p + . where C = M − − n (cid:0) n − n (cid:1) /p − and M is the norm of the maximal operator on L p ( · ) .In fact, we will proved this inequality with Γ min replaced by a sub-collection Γ L .Given a cube Q ∈ Γ , we say Q is lighted if | Light( Q ) | ≥ | Q | / n . Let Γ L bethe collection of lighted cubes. Observe that Γ L ⊂ Γ min . As was proved in [18,Lemma 4.3], for every finite set Γ of dyadic cubes,2 n − n card(Γ) ≤ card(Γ L ) ≤ card(Γ min ) ≤ card(Γ) . Hence, by Lemma 3.2, if Q ∈ Γ L , then k χ Light( Q ) k p ( · ) k χ Q k p ( · ) ≥ M | Light( Q ) || Q | ≥ n M . We can now estimate as follows: since p − ≤ p ( x ) ≤ p + , the sets Light( Q ), Q ∈ Γ L ,are disjoint and (2.1), Z R n C − N − /p + X Q ∈ Γ L χ Light( Q ) ( x ) k χ Q k p ( · ) ! p ( x ) dx = X Q ∈ Γ L Z Light( Q ) C − p ( x ) N − p ( x ) /p + k χ Q k − p ( x ) p ( · ) dx ≥ N X Q ∈ Γ L Z Light( Q ) C − p ( x ) k χ Q k − p ( x ) p ( · ) dx ≥ N X Q ∈ Γ L Z Light( Q ) C − p ( x ) n M ) p ( x ) k χ Light( Q ) k − p ( x ) p ( · ) dx = 1 N X Q ∈ Γ L Z Light( Q ) (cid:18) n − n (cid:19) − p ( x ) /p − k χ Light( Q ) k − p ( x ) p ( · ) dx ≥ n n − N X Q ∈ Γ L Z Light( Q ) k χ Light( Q ) k − p ( x ) p ( · ) dx = 2 n n − N card(Γ L ) ≥ N card(Γ) = 1 . REEDY BASES IN VARIABLE LEBESGUE SPACES 13
The desired inequality now follows by the definition of the L p ( · ) norm. (cid:3) To finish the proof of Theorem 1.1 we need to show that the bounds given inProposition 3.6 are sharp. This is an immediate consequence of the following result:since the constants in it are independent of ǫ , we can let ǫ → Proposition 3.7.
Given an exponent function p ( · ) , suppose < p − ≤ p + < ∞ and the Hardy-Littlewood maximal operator is bounded on L p ( · ) ( R n ) . Fix ǫ > and N ∈ N ; then there exists families Γ , Γ , of pairwise disjoint dyadic cubes such that (cid:13)(cid:13)(cid:13)(cid:13) X Q ∈ Γ χ Q k χ Q k p ( · ) (cid:13)(cid:13)(cid:13)(cid:13) p ( · ) ≥ C N p − + ǫ and (cid:13)(cid:13)(cid:13)(cid:13) X Q ∈ Γ χ Q k χ Q k p ( · ) (cid:13)(cid:13)(cid:13)(cid:13) p ( · ) ≤ C N p + − ǫ . Moreover, the constants C and C are independent of ǫ and N. Proof.
We first construct Γ . Let G ǫ = { x : p ( x ) ≤ p − + ǫ } . By the definition of p − , | G ǫ | >
0. Let x ∈ G ǫ be a Lebesgue point of the function χ G ǫ ; by the Lebesguedifferentiation theorem, if { Q k } is a sequence of dyadic cubes of decreasing side-lengthsuch that T k Q k = { x } , then lim k →∞ − Z Q k χ G ǫ ( x ) dx = 1 . Therefore, we can find a dyadic cube Q x containing x such that(3.4) | G ǫ ∩ Q x || Q x | ≥ . (Here, the choice of 1 / < c < Q x to be arbitrarily small.By fixing N such Lebesgue points, we can form a family Γ of disjoint cubes Q such that | Q ∩ G ǫ | > | Q | . By Lemma 3.2,(3.5) k χ G ǫ ∩ Q k p ( · ) k χ Q k p ( · ) ≥ M . (Again, M is the bound of the maximal operator on L p ( · ) .)We can now estimate as follows: since the cubes in Γ are disjoint and using (3.5), Z R n (cid:18) M N − p − + ǫ X Q ∈ Γ χ Q ( x ) k χ Q k p ( · ) (cid:19) p ( x ) dx = X Q ∈ Γ Z Q (2 M ) p ( x ) N − p ( x ) p − + ǫ k χ Q k − p ( x ) p ( · ) dx ≥ X Q ∈ Γ Z G ǫ ∩ Q N − p ( x ) p − + ǫ k χ G ǫ ∩ Q k − p ( x ) p ( · ) dx ≥ X Q ∈ Γ N − Z G ǫ ∩ Q k χ G ǫ ∩ Q k − p ( x ) p ( · ) dx = 1 . In the second inequality we use the fact that p ( x ) ≤ p − + ǫ a.e. in G ǫ and the lastinequality follows from (2.1). By the definition of the norm, this gives us the firstinequality with C = 2 M .The construction of Γ is similar but requires a more careful selection of Lebesguepoints. Let H ǫ = { x : p ( x ) ≥ p + − ǫ } ; again we have that | H ǫ | >
0. Let x be aLebesgue point of the function χ H ǫ contained in the set H ǫ/ and also such that x is a Lebesgue point of the locally integrable function p ( · ) − . Then by the Lebesguedifferentiation theorem we can find an arbitrarily small dyadic cube Q containing x such that(3.6) | H ǫ ∩ Q || Q | > − N .
Moreover, since (again by the Lebesgue differentiation theorem) − Z Q p ( y ) dy → p ( x ) ≤ p + − ǫ/ , we may also choose Q so small that(3.7) 1 p Q = − Z Q p ( y ) dy < p + − ǫ . Finally, choose N such Lebesgue points and take the cubes Q small enough that theyare pairwise disjoint and so that | Q | ≤
1. This gives us our family Γ .Fix a constant C >
1; the exact value will be determined below. We can nowestimate as follows: Z R n (cid:0) C N (cid:1) − p + − ǫ X Q ∈ Γ χ Q ( x ) k χ Q k p ( · ) ! p ( x ) dx = X Q ∈ Γ Z Q (cid:0) C N (cid:1) − p ( x ) p + − ǫ k χ Q k − p ( x ) p ( · ) dx = X Q ∈ Γ Z H ǫ ∩ Q + Z Q \ H ǫ = I + I . The estimate for I is immediate: since p ( x ) ≥ p + − ǫ in H ǫ we have that I ≤ C N X Q ∈ Γ Z H ǫ ∩ Q k χ H ǫ ∩ Q k − p ( x ) p ( · ) dx = 12 C < . REEDY BASES IN VARIABLE LEBESGUE SPACES 15
To estimate I , note that by Lemma 3.1, k χ Q k − p ( · ) ≤ | Q | − pQ . Then, since 2 N ≥ p − and p + we have that I ≤ X Q ∈ Γ Z Q \ H ǫ (2 N ) − p ( x ) p + − ǫ C − p ( x ) p + − ǫ p ( x ) | Q | − p ( x ) pQ dx ≤ p + C − p − p + − ǫ X Q ∈ Γ Z Q \ H ǫ | Q | − p ( x ) pQ dx . In Q \ H ǫ , p ( x ) < p + − ǫ < p Q by (3.7). Thus, | Q | − p ( x ) pQ < | Q | − since | Q | < | Q \ H ǫ || Q | < N . Hence, I ≤ p + C − p − p + − ǫ X Q ∈ Γ | Q \ H ǫ || Q | ≤ p + C − p − p + − ǫ N N = 12 , where the last equality holds if we choose C such that2 p + = C p − p + − ǫ . Since I + I ≤
1, again by the definition of the L p ( · ) norm we have that (cid:13)(cid:13)(cid:13)(cid:13) X Q ∈ Γ χ Q k χ Q k p ( · ) (cid:13)(cid:13)(cid:13)(cid:13) p ( · ) ≤ (cid:0) C N (cid:1) p + − ǫ = 2 p + − ǫ p + p − N p + − ǫ . This completes the proof of Proposition 3.7 with C = 2 p + p − +1 . (cid:3) Weighted Variable Lebesgue Spaces
We begin with some preliminary definitions and results on weighted variable Lebes-gue spaces. For proofs and further information, see [4]. Given an exponent p ( · ) wesay that a weight w ∈ A p ( · ) if[ w ] A p ( · ) = sup Q | Q | − k wχ Q k p ( · ) k w − χ Q k p ′ ( · ) < ∞ . This definition generalizes the Muckenhoupt A p classes to the variable setting. Wedefine the weighted variable Lebesgue space L p ( · ) ( w ) to the set of all measurablefunctions f such that k f w k p ( · ) < ∞ .If 1 < p − ≤ p + < ∞ , p ( · ) is log-H¨older continuous locally and at infinity, and w ∈ A p ( · ) , then the maximal operator is bounded on L p ( · ) ( w ): there exists a constant C such that k ( M f ) w k p ( · ) ≤ C k f w k p ( · ) . Note that with these hypotheses, we have that p ′ ( · ) is also log-H¨older continuous and w − ∈ A p ′ ( · ) ; thus, the maximal operator is also bounded on L p ′ ( · ) ( w − ). Because ofthis, we make the following definition: given an exponent p ( · ) and a weight w ∈ A p ( · ) ,we say that ( p ( · ) , w ) is an M -pair if the maximal operator is bounded on L p ( · ) ( w )and L p ′ ( · ) ( w − ). We can now state the analog of Theorem 1.1 for weighted variable Lebesgue spaces.
Theorem 4.1.
Given an exponent function p ( · ) , < p − ≤ p + < ∞ , and a weight w ∈ A p ( · ) , suppose ( p ( · ) , w ) is an M -pair. Let Ψ be an admissible orthonormal waveletfamily. The right and left democracy functions of Ψ in L p ( · ) ( w ) satisfy h r ( N ) ≈ N /p − , h l ( N ) ≈ N /p + , N = 1 , , , . . . Proof.
The proof of Theorem 4.1 is nearly identical to the proof of Theorem 1.1: herewe describe the changes.First, we need the analog of Theorem 2.1 for the weighted variable Lebesgue spaces.Theorem 2.1 was proved in [5] using Rubio de Francia extrapolation in the scale ofvariable Lebesgue spaces. Extrapolation can also be used to prove norm inequalitiesin the weighted space L p ( · ) ( w ) provided that ( p ( · ) , w ) is an M -pair: this was provedrecently in [6]. Therefore, the same proof as in [5] yields(4.1) k ( W Ψ f ) w k p ( · ) ≈ k f w k p ( · ) . We replace Lemma 3.1 with its weighted version:(4.2) W ( Q ) pQ,w ≤ k χ Q w k p ( · ) , where we set W ( x ) = w ( x ) p ( x ) and1 p Q,w = 1 W ( Q ) Z Q p ( x ) W ( x ) dx = − Z Q p ( x ) dW. The proof follows that of the unweighted version replacing dx by dW . Before usingH¨older’s inequality we divide and multiply by w and at the last step we use that k W ( Q ) − /p ′ ( · ) W w − k p ′ ( · ) = 1 by (2.1).The weighted versions of Lemmas 3.2 and 3.3 hold: | E || Q | ≤ M w k χ E w k p ( · ) k χ Q w k p ( · ) (4.3) k χ E w k p ( · ) k χ Q w k p ( · ) ≤ C (cid:18) | E || Q | (cid:19) δ , (4.4)where M w is the norm of the maximal operator on L p ( · ) ( w ). The proofs follow thesame steps, using the fact that since ( p ( · ) , w ) is an M -pair, the maximal operatoris bounded on L p ( · ) ( w ) and L p ′ ( · ) ( w − ). In the proof of (4.4) the following changesare required. First, we construct the Rubio de Francia iteration algorithm using thenorm of the maximal operator on L p ( · ) ( w − ) so that k ( Rg ) w − k p ( · ) ′ ≤ k g w − k p ( · ) ′ .Second, we replace Rg by R ( gw ). Third, before applying H¨older’s inequality wemultiply and divide by w .To modify the proof of Theorem 4.1 proper we use (4.1) to replace (3.1) with REEDY BASES IN VARIABLE LEBESGUE SPACES 17 (4.5) (cid:13)(cid:13)(cid:13)(cid:13) X Q ∈ Γ ψ Q w k ψ Q w k p ( · ) (cid:13)(cid:13)(cid:13)(cid:13) p ( · ) ≈ (cid:13)(cid:13)(cid:13)(cid:13) X Q ∈ Γ ψ Q w | Q | − / k χ Q w k p ( · ) (cid:13)(cid:13)(cid:13)(cid:13) p ( · ) ≈ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) X Q ∈ Γ χ Q k χ Q w k p ( · ) (cid:19) / w (cid:13)(cid:13)(cid:13)(cid:13) p ( · ) . The proof of the weighted version of Proposition 3.4 is exactly the same, replacing dx by dW and using the fact that by (2.1), for any set E ,(4.6) Z E k χ E w k − p ( x ) p ( · ) dW = Z E (cid:18) w ( x ) k χ E w k p ( · ) (cid:19) p ( x ) dx = 1 . The linearization estimate in Proposition 3.5 is the same, but defining S Γ ( x ) = X Q ∈ Γ χ Q k χ Q w k p ( · ) ! / , replacing dx by dW and using (4.4) instead of Lemma 3.3. The proof of Proposi-tion 3.6 is the same, replacing dx by dW and Lemma 3.2 by (4.3) and using (4.6):the properties of lighted and shaded cubes are geometric and so remain unchanged.Finally, the proof of Proposition 3.7 requires the following changes. We constructΓ much as before (in particular the Lebesgue differentiation theorem is used inexactly the same manner). The proof then proceeds the same way with dW in placeof dx , with (4.3) replacing Lemma 3.2 and by using at the end (4.6). To construct Γ we consider the same set H ǫ but now the Lebesgue differentiation theorem is appliedto p Q,w with respect to the measure dW and for dyadic cubes. (Recall that thedyadic Hardy-Littlewood maximal function defined with respect to the measure dW is of weak-type (1 ,
1) with respect to dW since 0 < W ( Q ) < ∞ for every dyadic cube Q .) In particular we obtain W ( H ǫ ∩ Q ) /W ( Q ) > − (2 N ) − and 1 /p Q,w < ( p + − ǫ ) − which we use to replace (3.6) and (3.7), respectively. Also, the cubes Q are taken sosmall that W ( Q ) ≤
1. Given these changes the remainder of the proof is the same mutatis mutandis , replacing dx by dW , p Q by p Q,w , Lemma 3.1 by (4.2), and usingagain (4.6). (cid:3) Variable Exponent Triebel-Lizorkin Spaces
The theory of (nonhomogeneous) Triebel-Lizorkin spaces with variable exponentshas been developed by Diening, et al. [10] and Kempka [25, 26]. (Also see Xu [36].)We refer the reader to these papers for complete information. Here, we sketch theessentials.Let P be the set of all measurable exponent functions p ( · ) : R n → (0 , ∞ ). Thenwith the same definitions and notation as used above, we can define the spaces L p ( · ) ; if p − <
1, then k · k p ( · ) is a quasi-norm and L p ( · ) is a quasi-Banach space. Themaximal operator will no longer be bounded on such spaces; a useful substitute isthe assumption that there exists p , 0 < p < p − , such that the maximal operatoris bounded on L p ( · ) /p ( R n ). This is the case if, for instance, if 0 < p − ≤ p + < ∞ and p ( · ) is log-H¨older continuous locally and at infinity. (For further information onthese spaces, see [7], where they were used to define variable Hardy spaces.)To define the variable exponent Triebel-Lizorkin spaces we need three exponentfunctions, p ( · ), q ( · ), and s ( · ). We let p ( · ) , q ( · ) ∈ P be such that 0 < p − ≤ p + < ∞ ,0 < q − ≤ q + < ∞ , and p ( · ) , q ( · ) are log-H¨older continuous locally and at infinity(see Section 2). We assume that s ( · ), the “smoothness” parameter, is in L ∞ and islocally log-H¨older continuous. (We note that in [10] it was assumed that s − ≥ F s ( · ) p ( · ) ,q ( · ) ( R n ) is defined using anapproximation of the identity on R n : for a precise definition, see [10, Definition 3.3]or [25, Section 4]. These spaces have many properties similar to those of the usual(constant exponent) Triebel-Lizorkin spaces. In particular, if 1 < p − ≤ p + < ∞ , F p ( · ) , ( R n ) = L p ( · ) ( R n ). For p − > F p ( · ) , ( R n ) = h p ( · ) ( R n ), the local Hardy spaceswith variable exponent introduced by Nakai and Sawano [31]. When s ≥ F sp ( · ) , ( R n ) = L s,p ( · ) ( R n ), the variable exponent Bessel potential spaces intro-duced in [1, 19]. When s ∈ N these become the variable exponent Sobolev spaces, W s,p ( · ) (see [3, Chapter 6]).A wavelet decomposition of variable exponent Triebel-Lizorkin spaces was provedin [26]. Let D + be the collection of all dyadic cubes Q such that | Q | ≤
1. Givenan orthonormal wavelet family Ψ = { ψ , ψ , · · · , ψ L } ⊂ L ( R n ) with appropri-ate smoothness and zero-moment conditions (determined by the exponent functions p ( · ) , q ( · ) , s ( · )) we have that f ∈ F s ( · ) p ( · ) ,q ( · ) if and only if(5.1) f = L X l =1 X Q ∈D + h f, ψ lQ i ψ lQ , and this series converges unconditionally in F s ( · ) p ( · ) ,q ( · ) . Moreover, if we define W s ( · ) ,q ( · )Ψ f ( x ) = L X l =1 X Q ∈D + (cid:16) |h f, ψ lQ i|| Q | − s ( x ) n − χ Q ( x ) (cid:17) q ( x ) q ( x ) , then(5.2) k f k F s ( · ) p ( · ) ,q ( · ) ≈ kW s ( · ) ,q ( · )Ψ f k p ( · ) . REEDY BASES IN VARIABLE LEBESGUE SPACES 19
We want to stress that the above result is only known for the nonhomogeneous,variable exponent Triebel-Lizorkin spaces, and it remains an open problem to defineand prove the basic properties of variable exponent Triebel-Lizorkin spaces in thehomogeneous case. (See [10, Remark 2.4].) Nevertheless, we can define the space˙ F s ( · ) p ( · ) ,q ( · ) with norm(5.3) k f k ˙ F s ( · ) p ( · ) ,q ( · ) = k ˙ W s ( · ) ,q ( · )Ψ f k p ( · ) . where we define ˙ W s ( · ) ,q ( · )Ψ exactly as in (5.1) except that the sum is taken over all Q ∈ D .The arguments given in Section 3 let us extend Theorem 1.1 to the variable ex-ponent Triebel-Lizorkin spaces. We first consider the homogeneous case ˙ F s ( · ) p ( · ) ,q ( · ) ( R n )with a constant smoothness parameter. Theorem 5.1.
Let p ( · ) , q ( · ) ∈ P be two exponent functions that are log-H¨oldercontinuous locally and at infinity and that satisfy < p − ≤ p + < ∞ , < q − ≤ q + < ∞ Let s ∈ R . Suppose that Ψ is an orthonormal wavelet family with sufficientsmoothness. Then the right and left democracy functions of Ψ in ˙ F sp ( · ) ,q ( · ) ( R n ) satisfy h r ( N ) ≈ N /p − , h l ( N ) ≈ N /p + , N = 1 , , , . . . Proof.
To modify the proof of Theorem 1.1 we must first give variants of Lemmas 3.1,3.2, and 3.3. Fix p , 0 < p < p − . Then, as we noted above, the maximal operator isbounded on L p ( · ) /p . Moreover, by a change of variable in the definition of the L p ( · ) norm, we have that for any set E ⊂ R n and τ > k χ E k p ( · ) = k χ τE k p ( · ) = k χ E k ττp ( · ) . Therefore, if we apply Lemma 3.1 to the exponent p ( · ) /p , we get that(5.4) | Q | p pQ ≤ k χ Q k p ( · ) /p = 2 k χ Q k p p ( · ) . Similarly, we can conclude from Lemmas 3.2 and 3.3 that if E ⊂ Q , then | E || Q | ≤ M (cid:18) k χ E k p ( · ) k χ Q k p ( · ) (cid:19) p (5.5)and (cid:18) k χ E k p ( · ) k χ Q k p ( · ) (cid:19) p ≤ C (cid:18) | E || Q | (cid:19) δ . (5.6) Turning to the proof proper, we may first assume, as in the proof of Theorem 1.1,that L = 1. We then need to prove the lower and upper bounds for (cid:13)(cid:13)(cid:13)(cid:13) X Q ∈ Γ ψ Q k ψ Q k ˙ F sp ( · ) ,q ( · ) (cid:13)(cid:13)(cid:13)(cid:13) ˙ F sp ( · ) ,q ( · ) , where Γ is a finite set of dyadic cubes with card(Γ) = N . By (5.3),(5.7) k ψ Q k ˙ F sp ( · ) ,q ( · ) = | Q | − sn − k χ Q k p ( · ) , and so(5.8) (cid:13)(cid:13)(cid:13)(cid:13) X Q ∈ Γ ψ Q k ψ Q k ˙ F sp ( · ) ,q ( · ) (cid:13)(cid:13)(cid:13)(cid:13) ˙ F sp ( · ) ,q ( · ) = (cid:13)(cid:13)(cid:13)(cid:13) X Q ∈ Γ ψ Q | Q | − sn − k χ Q k p ( · ) (cid:13)(cid:13)(cid:13)(cid:13) ˙ F sp ( · ) ,q ( · ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) X Q ∈ Γ χ Q ( x ) k χ Q k q ( x ) p ( · ) (cid:19) /q ( x ) (cid:13)(cid:13)(cid:13)(cid:13) p ( · ) . When the cubes in Γ are pairwise disjoint, the proof of Proposition 3.4 is un-changed. To modify the proof of Proposition 3.5, define S p ( · ) ,q ( · )Γ ( x ) = (cid:18) X Q ∈ Γ χ Q ( x ) k χ Q k q ( x ) p ( · ) (cid:19) /q ( x ) . Then(5.9) (cid:13)(cid:13)(cid:13)(cid:13) X Q ∈ Γ ψ Q k ψ Q k ˙ F sp ( · ) ,q ( · ) (cid:13)(cid:13)(cid:13)(cid:13) ˙ F sp ( · ) ,q ( · ) = k S p ( · ) ,q ( · )Γ k p ( · ) . With the same notation as before, we clearly have that χ Q x ( x ) k χ Q x k p ( · ) ≤ S p ( · ) ,q ( · )Γ ( x ) . We prove the opposite inequality almost as before, using (5.6) instead of Lemma 3.3: S p ( · ) ,q ( · )Γ ( x ) ≤ (cid:18) X Q ⊃ Q x k χ Q k q ( x ) p ( · ) (cid:19) /q ( x ) ≤ (cid:18) ∞ X j =0 C k χ Q x k q ( x ) p ( · ) − jnδq ( x ) /p (cid:19) /q ( x ) = C χ Q x ( x ) k χ Q x k p ( · ) ;in the last inequality we use the fact that q ( x ) ≥ q − > . Therefore, S p ( · ) ,q ( · )Γ ( x ) ≈ χ Q x ( x ) k χ Q x k p ( · ) . REEDY BASES IN VARIABLE LEBESGUE SPACES 21
From here, the proof of Theorem 5.1 is exactly the same as that of Theorem 1.1:the proofs of Propositions 3.6 and 3.7 are the same since S Γ ( x ) ≈ S p ( · ) ,q ( · )Γ ( x ). We onlynote that because we use (5.4) in place of Lemma 3.1 and (5.5) instead of Lemma 3.2,some of the constants which appear must be adjusted to account for the exponent p . (cid:3) In the nonhomogeneous case we may take s ( · ) to be variable. Theorem 5.2.
Let p ( · ) , q ( · ) ∈ P be two exponent functions that are log-H¨oldercontinuous locally and at infinity and that satisfy < p − ≤ p + < ∞ , < q − ≤ q + < ∞ Let s ( · ) ∈ L ∞ be locally log-H¨older continuous. Suppose that Ψ is an orthonormalwavelet family with sufficient smoothness (i.e., so that (5.1) and (5.2) hold). Thenthe right and left democracy functions of Ψ in F s ( · ) p ( · ) ,q ( · ) ( R n ) satisfy h r ( N ) ≈ N /p − , h l ( N ) ≈ N /p + , N = 1 , , , . . . Proof.
The proof is nearly identical to the proof of Theorem 5.1. The key differenceis in equalities (5.7) and (5.8). In (5.7) we used the fact that s was constant in orderto pull the term | Q | − sn − out of the L p ( · ) norm. We can no longer do this if s ( · ) is afunction.However, we can use local log-H¨older continuity and the fact that | Q | ≤ s ( · ) is that there exists C > Q , | Q | s − ( Q ) − s + ( Q ) ≤ C. (See [3, Lemma 3.24].) In particular, for any x ∈ Q with | Q | ≤ | Q | − s ( x ) = | Q | − s ( x )+ s − ( Q ) | Q | − s − ( Q ) ≤ C | Q | − s − ( Q ) , | Q | − s ( x ) = | Q | − s ( x )+ s + ( Q ) | Q | − s + ( Q ) ≥ C − | Q | − s + ( Q ) . Therefore, by (5.2) we have that k ψ Q k F s ( · ) p ( · ) ,q ( · ) ≈ k| Q | − s ( · ) n − χ Q k p ( · ) . | Q | − s − ( Q ) n − k χ Q k p ( · ) and k ψ Q k F s ( · ) p ( · ) ,q ( · ) & | Q | − s +( Q ) n − k χ Q k p ( · ) . Moreover, because every Q ∈ Γ is such that | Q | ≤
1, we have that (cid:13)(cid:13)(cid:13)(cid:13) X Q ∈ Γ ψ Q k ψ Q k F s ( · ) p ( · ) ,q ( · ) (cid:13)(cid:13)(cid:13)(cid:13) F s ( · ) p ( · ) ,q ( · ) . (cid:13)(cid:13)(cid:13)(cid:13) X Q ∈ Γ ψ Q | Q | − s +( Q ) n − k χ Q k p ( · ) (cid:13)(cid:13)(cid:13)(cid:13) F s ( · ) p ( · ) ,q ( · ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) X Q ∈ Γ (cid:20) | Q | − s ( x ) n − χ Q ( x ) | Q | − s +( Q ) n − k χ Q k p ( · ) (cid:21) q ( x ) (cid:19) /q ( x ) (cid:13)(cid:13)(cid:13)(cid:13) p ( · ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) X Q ∈ Γ χ Q ( x ) k χ Q k q ( x ) p ( · ) (cid:19) /q ( x ) (cid:13)(cid:13)(cid:13)(cid:13) p ( · ) . In the same way, and again using strongly that | Q | ≤
1, we have (cid:13)(cid:13)(cid:13)(cid:13) X Q ∈ Γ ψ Q k ψ Q k F s ( · ) p ( · ) ,q ( · ) (cid:13)(cid:13)(cid:13)(cid:13) F s ( · ) p ( · ) ,q ( · ) & (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) X Q ∈ Γ χ Q ( x ) k χ Q k q ( x ) p ( · ) (cid:19) /q ( x ) (cid:13)(cid:13)(cid:13)(cid:13) p ( · ) . Given this equivalence, the proof now continues exactly as in the proof of Theo-rem 5.1. (cid:3)
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David Cruz-Uribe, SFO, Dept. of Mathematics, Trinity College, Hartford, CT06106-3100, USA
E-mail address : [email protected] Eugenio Hern´andez, Departamento de Matem´aticas, Universidad Aut´onoma deMadrid, E-28049 Madrid, Spain
E-mail address : [email protected] Jos´e Mar´ıa Martell, Instituto de Ciencias Matem´aticas CSIC-UAM-UC3M-UCM,Consejo Superior de Investigaciones Cient´ıficas, C/ Nicol´as Cabrera, 13-15, E-28049Madrid, Spain
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