An atomic decomposition of variable Besov and Triebel-Lizorkin spaces is given

Jingshi Xu

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Armenian Journal of MathematicsVolume 2, Number 1, 2009, 1–12An atomic decomposition of variable Besov andTriebel-Lizorkin spacesJingshi Xu** Department of Mathematics, Hunan Normal UniversityChangsha 410081, Peoples’s Republic of Chinajshixu@yahoo.com.cnAbstractAn atomic decomposition of variable Besov and Triebel-Lizorkin spaces is given.Key Words: variable exponent, Besov space, Triebel-Lizorkin space, atomic decomposition,Maximal operatorMathematics Subject Classification 2000: 46E35, 42B15IntroductionLet p be a measurable function on Rn with range in [1,∞). Lp(·)(Rn) denotes the set ofall measurable functions f on Rn such that for some λ > 0,∫Rn(|f(x)|λ)p(x)dx <∞.The set becomes a Banach function space when equipped with the norm‖f‖Lp(·) = inf{λ > 0 :∫Rn(|f(x)|λ)p(x)dx ≤ 1}.These spaces are referred to as variable Lebesgue spaces, since they generalized the standardLebesgue spaces. Note that one can define variable Lebesgue spaces on any measurablesubset of Rn, see [16]. However, in this paper we only work on the whole space Rn.Denote P(Rn) the set of all measurable functions p on Rn with range in [1,∞) such that1 < p− = ess infx∈Rnp(x), ess supx∈Rnp(x) = p+ <∞.1In the classical Lebesgue spaces one can work with Lp where 0 < p < 1. In this paper,we also consider analogous spaces with variable exponents. Define P0(Rn) to be the set ofall measurable functions p on Rn with range in (0,∞) such thatp− = ess infx∈Rnp(x) > 0, ess supx∈Rnp(x) = p+ <∞.Given p(·) ∈ P0(Rn), one can define the space Lp(·)(Rn) as above. This is equivalent todefining it to be the set of all functions f such that |f |p0 ∈ Lq(·)(Rn), where 0 < p0 < p−,and q(x) = p(x)p0∈P(Rn). Then one can define a quasi-norm on this space by‖f‖Lp(·) = ‖|f |p0‖1/p0Lq(·).Let f ∈ L1loc(Rn). Then the standard Hardy-Littlewood maximal function of f was definedbyMf(x) = supB3x1|B|∫B|f(y)|dy,where B denotes balls in Rn, and |B| is the volume of ball B.It is well known that the boundedness of the Hardy-Littlewood maximal operator onLebesgue spaces plays a key role in classical analysis. So does it on variable exponentLebesgue spaces. Let B(Rn) be the set of all p(·) ∈P(Rn) such that the Hardy-Littlewoodmaximal operator M is bounded on Lp(·)(Rn). There are some sufficient conditions on p(·)for p(·) ∈ B(Rn), see examples [5, 6, 8, 17, 18]. If the maximal operator M is bounded onspace Lp(·)(Rn), then many classical operators such as singular integrals and commutatorsare also bounded on the same space Lp(·)(Rn), see [4] and references therein.In recent decades, many attention has payed to the study of variable Lebesgue spaces,the corresponding variable Sobolev spaces W k,p(·)(Rn) and variable Bessel potential spacesLs,p(·)(Rn) with p(·) ∈ P(Rn). In fact, likewise to the classical situation, variable Besselpotential spaces Lm,p(·)(Rn) coincide to variable Sobolev spaces Wm,p(·)(Rn) when p(·) ∈B(Rn) and m is any integer. This result was obtained firstly by Almeida and Samko in [3]and again by Gurka, Harjulehto and Nekvinda in [12]. Moreover, these spaces have beenapplied to partial differential equations and the calculus of variation, see [1, 3, 4, 5, 6, 7, 8,10, 11, 12, 16, 17, 18, 19, 21, 22, 29] and references therein.It is well known that Besov and Triebel-Lizorkin spaces include many classical spacesas special cases, for example, the Hölder spaces, the Sobolev spaces, the Bessel potentialspaces, the Zygmund spaces, the local Hardy spaces and the space bmo(Rn). All the abovementioned spaces have been studied intensively and applied in many fields of analysis, suchas ordinary and partial differential equations; see for examples, [2, 20, 23, 24, 25, 26].Inspired by the mentioned references, similar to classical Besov and Triebel-Lizorkinspaces, the author introduced the variable Besov spaces and Triebel-Lizorkin spaces in [27].In fact, the author obtained a characterization of these new spaces by maximal operator.Then in [28], the author proved that if p(·) ∈ B(Rn) then the variable Bessel potential2spaces Ls,p(·)(Rn) and variable Triebel-Lizorkin space F sp(·),2(Rn) are coincident for s ∈ R, arelationship similar to the classical setting.It is known that atomic decomposition is an important property for classical functionspaces. In this paper, we will consider the atomic decomposition of the variable Besov andTriebel-Lizorkin spaces. Our results will be stated in the next section. At the end of thissection we recall the definition of these spaces.Let S (Rn) be the Schwartz space of all complex-valued rapidly decreasing infinitelydifferentiable functions on Rn. Let S ′(Rn) be the set of all the tempered distribution on Rn.If ϕ ∈ S(Rn), then ϕ̂ denotes the Fourier transform of ϕ, and ϕ∨ denotes the inverse Fouriertransform of ϕ. Let Φ ∈ S(Rn) satisfy the following conditions:supp Φ̂ ⊂ B(0, 1), and supp Φ̂ = 1 on B(0, 1/2).Set Φj(x) = 2njΦ(2jx), x ∈ Rn, for j ∈ Zn. We also putθj = Φj(x)− Φj−1(x).Denote θ0 = Φ. It follows that∞∑j=0θ̂j(ξ) ≡ 1.Definition 1 Let s ∈ R, 0 < q ≤ ∞, p(·) ∈ P0(Rn). Suppose θj, j ∈ N0 = N⋃{0}as above, then the variable exponent Triebel-Lizorkin space F sp(·),q(Rn) is the collection off ∈ S ′(Rn) such that ‖f‖F sp(·),q<∞, where the norm of f is‖f‖F sp(·),q=∥∥{2sjθj ∗ f}∞0 ∥∥Lp(·)(`q) .Similarly, the variable exponent Besov spaces Bsp(·),q(Rn) is the collection of f ∈ S ′(Rn) suchthat ‖f‖Bsp(·),q<∞, where the norm of f in this space is‖f‖Bsp(·),q=∥∥{2sjθj ∗ f}∞0 ∥∥`q(Lp(·)) .Here Lp(·)(`q) and `q(Lp(·)) are the spaces of all sequences {gj} of measurable functionson Rn with finite quasi-norms‖{gj}‖Lp(·)(`q) = ‖‖{gj}‖`q‖Lp(·) =∥∥∥∥∥∥(∞∑j=1|gj(x)|q) 1qdx∥∥∥∥∥∥Lp(·),and‖{gj}‖`q(Lp(·)) = ‖‖{gj}‖Lp(·)‖`q =(∞∑j=1‖gj‖qLp(·)) 1q.Throughout, the letter C denotes positive constants, but it may change line to line.Constants may in general depend on all fixed parameters, and sometimes we show thisdependence explicitly by writing, e.g. CN .31. Atomic decompositionSetxjk = 2−jk for j ∈ N0 and k ∈ Zn,then we define dyadic cubes Qjk byQjk = xjk + 2−j[0, 1)n.The characteristic function of Qjk is denoted by χjk.For any cube Q, its side is denoted by l(Q), and for λ > 0, λQ denotes the cube concentricto Q with side λl(Q). Denote 3Qjk by Q̃jk, and χ̃jk denotes the characteristic function forQ̃jk.Definition 2 Let S, T be a nonnegative integer. A function ajk in CS(Rn) is called a (S, T )atom for a cube Qjk if it satisfies the following conditions:supp ajk ⊂ Q̃jk;supx2−j|γ||Dγajk(x)| ≤ 1 for all |γ| ≤ S;if j > 0,∫Rnxγajk(x)dx = 0 for all |γ| ≤ T.Now we can state our first result.Theorem 1 Let p(·) ∈P0(Rn) with 0 < p0 < p− such that p(·)/p0 ∈ B(Rn).(i) Let f belong to Bsp(·),q then there exist (S, T ) atoms {ajk} for the dyadic {Qjk}, andcoefficients {tjk} such thatf =∞∑j=02−jsuj in S′ with uj =∑k∈Zntjkajk,and denoting tj =∑k∈Zn tjkχjk, there is a constant C independent of f such that‖{tj}∞0 ‖`q(Lp(·)) =(∞∑j=0∥∥∥∥∥∑k∈Zn|tjk|χjk∥∥∥∥∥qLp(·))1/q≤ C‖f‖Bsp(·),q. (1)(ii) Let f belong to F sp(·),q then there exist (S, T ) atoms {ajk} for the dyadic {Qjk}, andcoefficients {tjk} such thatf =∞∑j=02−jsuj in S′ with uj =∑k∈Zntjkajk, (2)and denoting tj =∑k∈Zn tjkχjk, there is a constant C independent of f such that‖{tj}∞0 ‖Lp(·)(`q) =∥∥∥∥∥∥(∞∑j=0∑k∈Zn|tjkχjk|q)1/q∥∥∥∥∥∥Lp(·)≤ C‖f‖F sp(·),q.4To prove Theorem 1, we follow the method used in the classical setting in [13, 14, 15].Firstly, we need a preliminary.The following lemma is the estimate for vector-valued setting in variable Lebesgue spaces,one can see Corollary 2.1 in [4].Lemma 1 If p(·) ∈ B(Rn), then for all 1 < q ≤ ∞,‖{Mfj}‖Lp(·)(`q) ≤ C‖{fj}‖Lp(·)(`q),where M is the Hardy-Littlewood maximal operator.For convenience, we use the notation Mr(g) = (M(|g|r))1/r.Proof of Theorem 1. From the notation in last section, we have thatf =∞∑j=0θj ∗ f =∞∑j=02−jsuj,where uj = 2jsθj ∗ f for j ∈ N0.Let η ∈ C∞0 (Q00),∫η(x)dx = 1, ηj(x) = 2jnη(2jx). Set ηjk = ηj ∗χjk, so that supp ηjk ⊂Q̃jk and∑k∈Zn ηjk ≡ 1.Set bjk = ηjkuj,tjk = maxx∈Q̃jk,|γ|≤S2−n|γ||Dγbjk(x)|,andajk =bjktjk,of course if tjk = 0, then ajk = 0.Thenf =∞∑j=02−js∑k∈Zntjkajk.Observe that Φ̂j+1(ξ) = 1 on supp ûj, so that uj = Φj+1 ∗ uj. Since for x ∈ Q̃jk andz ∈ Qjk, there exist constants C1, C2 independent of j such that 0 < C1 ≤ 1+2j |x−y|1+2j |z−y| ≤ C25for y ∈ Rn, then for |γ| ≤ S and z ∈ Qjk, we have|Dγuj(x)| ≤ |DγΦj+1| ∗ |uj|(x)=∫Rn|DγΦj+1(x− y)||uj(y)|dy≤ C2j|γ|2jn∫Rn|uj(y)|(1 + 2j|x− y|)Ldy≤ C2j|γ|2jn∫Rn|uj(y)|(1 + 2j|z − y|)Ldy≤ C2j|γ|2jn∫Rn|uj(y)|(1 + 2j|z − y|)n/r1(1 + 2j|z − y|)L−n/rdy≤ C2j|γ|Mr(uj)(z)∫Rn2jn(1 + 2j|z − y|)L−n/rdy= C2j|γ|Mr(uj)(z)∫Rn1(1 + |y|)L−n/rdy≤ C2j|γ|Mr(uj)(z)∫ ∞01(1 + t)L−n/r−n+1dt≤ C2j|γ|Mr(uj)(z)≤ C2j|γ| infz∈Qjk Mr(uj)(z),where we supposed L > n+ n/r, and we usedsupy∈Rn|uj(y)|(1 + 2j|z − y|)n/r≤ CMr(uj)(z) (3)for uj ∈ S ′(Rn) such that supp ûj ⊂ B(0, 2j). We leave the proof of (3) to the end of theproof.By the Leibniz’s formulatjk ≤ C maxx∈Q̃jk,|γ|≤S2−j|γ||Dγuj(x)|.Therefore tjk ≤ CMr(uj)(x) for all x ∈ Qjk, and thustj(x) =∑k∈Zntjkχjk ≤ CMr(uj)(x).Now the inequalities (1) and (2) follow by the boundedness of the Hardy-Littlewoodoperator and its vector-valued operator, Lemma 1. Let us check (2). Choose r such thatr < min{q, p0} (for (1), only 0 < r < p0), by Theorem 8.1 in [8], p(·)/r ∈ B(Rn). Hence, by6Lemma 1 we have‖{tj}∞0 ‖Lp(·)(`q) =∥∥∥∥∥∥(∞∑j=0∑k∈Zn|tjkχjk|q)1/q∥∥∥∥∥∥Lp(·)≤ C∥∥∥∥∥∥(∞∑j=0|Mruj|q)1/q∥∥∥∥∥∥Lp(·)= C∥∥∥∥∥∥(∞∑j=0|M |uj|r|q/r)r/q∥∥∥∥∥∥1/rLp(·)/r≤ C∥∥∥∥∥∥(∞∑j=0|uj|q)1/q∥∥∥∥∥∥Lp(·)= C‖f‖F sp(·),q.Finally, we show (3) is true. From Theorem 1.3.1 on page 16 in [23], we know that for0 < r <∞supy∈Rn|φ(x− y)|(1 + |y|)n/r≤ CMr(φ)(x)holds for all x ∈ Rn and φ ∈ S such that supp φ̂ ⊂ B(0, 1). Then with the argument in Step1 of the proof of Theorem 1.4.1 on page 22 in [23],supy∈Rn|φ(x− y)|(1 + |y|)n/r≤ CMr(φ)(x)holds for all x ∈ Rn and φ ∈ S ′(Rn) such that supp φ̂ ⊂ B(0, 1). Thus by scaling andtranslation,supy∈Rn|φ(x− y)|(1 + 2j|y|)n/r≤ CMr(φ)(x)holds for all x ∈ Rn and φ ∈ S ′(Rn) such that supp φ̂ ⊂ B(0, 2j). In other words,|φ(y)|(1 + 2j|y − x|)n/r≤ CMr(φ)(x)holds for all x, y ∈ Rn and φ ∈ S ′(Rn) such that supp φ̂ ⊂ B(0, 2j). This is (3).This completes the proof.Conversely, we haveTheorem 2 Let p(·) ∈P0(Rn) with 0 < p0 < p− such that p(·)/p0 ∈ B(Rn).(i) Let {amk} be a sequence (S, T ) atoms for the dyadic {Qmk}m∈N0,k∈Zn , where S >s, T > −1− n− s+ n/min{p0, 1}, and let coefficients {tmk} such that‖{tm}∞0 ‖`q(Lp(·)) =(∞∑m=0∥∥∥∥∥∑k∈Zn|tmk|χmk∥∥∥∥∥qLp(·))1/q<∞,7where tm =∑k∈Zn tmkχmk. Then the function f defined in S′(Rn) byf =∞∑m=02−ms∑k∈Zntmkamkbelongs to Bsp(·),q, and there is a constant C independent of f such that‖f‖Bsp(·),q≤ C‖{tm}∞0 ‖`q(Lp(·)) = C(∞∑m=0∥∥∥∥∥∑k∈Zn|tmk|χmk∥∥∥∥∥qLp(·))1/q.(ii) Let {amk} be a sequence (S, T ) atoms for the dyadic {Qmk}m∈N0,k∈Zn , where S >s, T > −1− n− s+ n/min{p0, q, 1}, and let coefficients {tmk} such that‖{tm}∞0 ‖Lp(·)(`q) =∥∥∥∥∥∥(∞∑m=0∑k∈Zn|tmk|qχmk)1/q∥∥∥∥∥∥Lp(·)<∞,where tm =∑k∈Zn tmkχmk. Then the function f defined in S′(Rn) byf =∞∑m=02−ms∑k∈Zntmkamkbelongs to F sp(·),q, and there is a constant C independent of f such that‖f‖F sp(·),q≤ C‖{tm}∞0 ‖Lp(·)(`q) = C∥∥∥∥∥∥(∞∑m=0∑k∈Zn|tmk|qχmk)1/q∥∥∥∥∥∥Lp(·).To prove Theorem 2, we require the following two lemmas, for them, see [13], [14].Lemma 2 Let {θj}∞0 as in Section 1 and let amk be a (S, T ) atom for Qmk. Then for x ∈ Rn|θj ∗ amk(x)| ≤C2−(m−j)T(1 + 2j|x− xmk|)Lfor 0 ≤ j ≤ m,and|θj ∗ amk(x)| ≤C2−(j−m)S(1 + 2m|x− xmk|)Lfor 0 ≤ m ≤ j,where L is sufficiently large, for our purpose, L > n/min{1, p−, q}.Lemma 3 Let 0 < r ≤ 1, L > n/r, and η ≥ 0. For any sequence {tjk}j∈N0,k∈Zn, there existsa constant C such that∑j=µ,k∈Zn|tjk|(1 + 2µ−η|x− xjk|)L≤ C2ηn/rMr( ∑j=µ,k∈Zn|tjk|χQjk)(x), x ∈ Rn.8Now we show Theorem 2.Proof Let f(x) =∞∑m=02−ms∑k∈Zntmkamk(x). By Lemma 2 and 3,2js|θj ∗ f(x)| ≤j∑m=02(j−m)s∑k∈Zn|tmk||θj ∗ amk|+∞∑m=j+12(j−m)s∑k∈Zn|tmk||θj ∗ amk|≤ Cj∑m=02−(j−m)(S−s)∑k∈Zn|tmk|(1 + 2m|x− xmk|)L+C∞∑m=j+12−(m−j)(T+1+n+s)∑k∈Zn|tmk|(1 + 2j|x− xmk|)L≤ Cj∑m=02−(j−m)(S−s)Mr(tm)(x)+C∞∑m=j+12−(m−j)(T+1+n+s−n/r)Mr(tm)(x),(4)where in the last inequality, we used η = 0 in Lemma 3 for the first part and η = m− j forthe second part.Thus, by the assumption, if we choose 0 < r < min{p0, 1} such that T+1+n+s−n/r > 0,then by Theorem 8.1 in [8], p(·)/r ∈ B(Rn). Therefore by Lemma 1 we obtain2js‖θj ∗ f‖Lp(·) ≤ Cj∑m=02−(j−m)(S−s)‖Mr(tm)‖Lp(·)+C∞∑m=j+12−(m−j)(T+1+n+s−n/r)‖Mr(tm)‖Lp(·)≤ Cj∑m=02−(j−m)(S−s)‖M(|tm|r)‖1/rLp(·)/r+C∞∑m=j+12−(m−j)(T+1+n+s−n/r)‖M(|tm|r)‖1/rLp(·)/r≤ Cj∑m=02−(j−m)(S−s)‖tm‖Lp(·)+C∞∑m=j+12−(m−j)(T+1+n+s−n/r)‖tm‖Lp(·)= C∞∑m=0Cj−m‖tm‖Lp(·) ,9and∑∞m=−∞Cm <∞. Using Minkowski’s inequality ‖a ∗ b‖`q ≤ ‖a‖`1‖b‖`q , we obtain that(∞∑j=0‖2jsθj ∗ f‖qLp(·))1/q≤ C(∞∑m=0‖tm‖qLp(·))1/q.This is (i).For (ii), let us pick 0 < r < min{p0, q, 1} such that T + 1 + n+ s− n/r > 0, from (4) byMinkowski’s inequality again, we have(∞∑j=0|2jsθj ∗ f(x)|q)1/q≤ C(∞∑m=0|Mr(tm)(x)|q)1/q.By Theorem 8.1 in [8] again, we have p(·)/r ∈ B(Rn). Thus, by Lemma 1 we have‖{2jsθj ∗ f}∞j=0‖Lp(·)(`q) ≤ C‖{Mr(tm)}∞m=0‖Lp(·)(`q)= C‖{M(|tm|r)}‖1/rLp(·)/r(`q/r)≤ C‖{|tm|}‖Lp(·)(`q)..This finishes the proof.Remark When this manuscript was finished, the author received the preprint [9] byProfessor Hästö. In [9], Triebel-Lizorkin spaces with variable smoothness and integrabilitywere introduced, which include the variable Triebel-Lizorkin spaces while s > 0 in thispaper as a special case, but do not include those for s < 0. In fact, atomic and moleculedecompositions and applications of these new Triebel-Lizorkin spaces were gave, for detail,see [9]. However, the variable Besov spaces in this paper was not considered in [9], thus theauthor presented these results here.AcknowledgmentsThe author would like to thank the referee for his careful reading which made the manuscriptmore readable. When this manuscript was written the author was visiting Karlsruhe Uni-versity. He is deeply grateful to Professor Lutz Weis and Department of Mathematics ofKarlsruhe University for their hospitality. 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An atomic decomposition of variable Besov and Triebel-Lizorkin spaces

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