Research Article Local Characterizations of Besov and Triebel-Lizorkin Spaces with Variable Exponent

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1 Function Spaces, Article ID , 8 pages Research Article Local Characterizations of Besov and Triebel-Lizorkin Spaces with Variable Exponent Baohua Dong and Jingshi Xu Department of Mathematics, Hainan Normal University, Haikou , China Correspondence should be addressed to Jingshi Xu; jingshixu@126com Received 29 May 2013; Accepted 30 October 2013; Published 22 January 2014 Academic Editor: Kehe Zhu Copyright 2014 B Dong and J Xu This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited We introduce new Besov and Triebel-Lizorkin spaces with variable integrable exponent, which are different from those introduced by the second author early Then we characterize these spaces by the boundedness of the local Hardy-Littlewood maximal operator on variable exponent Lebesgue space Finally the completeness and the lifting property of these spaces are also given 1 Introduction Variable exponent function spaces have attracted many attentions because of their applications in some aspects, such as partial differential euations with nonstandard growth [1], electrorheological fluids [2], and image restoration [3 5] In fact, since the variable Lebesgue and Sobolev spaces were systemically studied by Kováčik and Rákosník in [6], there are many spaces introduced, such as, Bessel potential spaces with variable exponent, Besov and Triebel-Lizorkin spaces with variable exponents, Morrey spaces with variable exponents, and Hardy spaces with variable exponent; see [7 20] and references therein When the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue spaces, many results in classical harmonic analysis and function theory hold for the variable exponent case; see [21 23] Let p : R n [1,beameasurable function Denote by L p R n thespaceofallmeasurablefunctionsf on R n such that for some λ>0 px f x dx <, 1 R n λ with the norm f L p R n := { R n px f x dx 1} 2 λ Then L p R n is a Banach space with the norm L p R n We will use the following notations: p := ess inf{px : x R n } and p + := ess sup{px : x R n }ThesetPR n consists of all p satisfying p >1and p + <Moreover, we define P 0 R n to be the set of measurable functions p on R n with the range in 0, such that 0<p p + < Given p P 0 R n, one can define the space L p R n as above This is euivalent to defining it to be the set of all functions f such that f p 0 L p /p 0 R n,where0<p 0 <p and p /p 0 PR n We also define a uasinorm on this space by f L p R n := f p 0 1/p 0 L p /p 0 R n Let f be a locally integrable function on R n ;thelocal variant of the Hardy-Littlewood maximal operator is given by 1 M loc f x := sup x, C f y dy, x Rn, 3 for some constant C We denote B loc R n the set of p PR n such that M loc is bounded on L p R n In2013 Danelia et al gave characterizations of B loc R n,avectorestimate for the local Hardy-Littlewood maximal operator if p B loc R n, and a Littlewood-Paley suare-function characterization of the variable exponent Lebesgue spaces L p R n when p belongs to B loc R n in [24] In 2001 Rychkov used the boundedness of the local Hardy-Littlewood maximal operator to prove a stronger result of thepeetretypeforspacesf s,ω p, Rn and B s,ω p, Rn andgavethe lifting property for these spaces in [25]

2 2 Function Spaces Motivated by the previous papers, the goal of this paper is to introduce new Besov and Triebel-Lizorkin spaces with variable exponent To state our result, we need some notations Throughout this paper S denotes the Lebesgue measure for a measurable set S R n N 0 denotes the set of all nonnegative integers Let D := C 0 Rn and D be the dual space of DFors R, s + := max{s, 0} and [s] is the largest integer less than or eual to s Given a function φ on R n,letl φ N 0 denote the maximal number such that φ has vanishing moments up to order L φ In other words, R n x α φxdx =0forallmultiindices α with α L φ If no moments of φ vanish, then put L φ = 1Apair of functions φ 0,φ is called satisfying the M s condition, if R n φ 0 xdx =0and L φ [s] Take a function φ 0 D satisfying M s condition, which is possible for any s Indeed, the assumption L φ [s]is void for s<0andissatisfied automatically for 0 s<1fors 1, any φ 0 D with Fourier transform φ 0 ξ = 1 + O ξ [s]+1 neartheoriginwilldothejobthenotations e Rn was introduced by Schott in [26] More precisely, let it be the set of all f D for which the estimate f, γ Asup { Dα γ x exp N x :x Rn, α N}, all γ D, 4 isvalidwithsomeconstantsa=a f, N=N f It is evidently that S e Rn includes temperate distributions S R n Now, we give the definition of Besov spaces and Triebel- Lizorkin spaces with variable exponent Definition 1 Let p P 0 R n, s R, 0<, φ 0 as above, φx := φ 0 x 2 n φ 0 x/2, andφ j x := 2 jn φ2 j x for j N i The Besov space with variable exponent B s, p Rn is the set of f with where {f S e Rn : f B s, p Rn <}, 5 f B s, p Rn := 2 js φ j f 1/ L p R n 6 ii For p + <, the Triebel-Lizorkin space with variable exponent F s, p Rn is the set of f with where {f S e Rn : f F s, p Rn <}, 7 f F s, p Rn := 2 js φ j f 1/Lp Rn 8 The key point is to prove that different choices of φ 0 in Definition 1 do not really change the spaces, leading to euivalent uasinorms For f S R n that has been proved by the second author in 2008, see [19] To go on, we recall variant Peetre-type maximal functions which was introduced by Rychkov in [25] Let φ j,a,b where f x := sup y R n φ j fx y, x R n, j N m j,a,b y 0, 9 m j,a,b y := j y A 2 y B, A,B 0 10 Now it is the position to state our main result Theorem 2 Let s R, 0<<,andp P 0 R n with 0<p 0 < min{p,}such that p /p 0 B loc R n Suppose that φ 0, ζ 0 D and the pairs φ 0,φ := φ 0 2 n φ 0 /2, and ζ 0,ζ := ζ 0 2 n ζ 0 /2 satisfy the M s condition Then there arepositiveconstantsa 0 := A 0 s,, B 0 := B 0 n,andc such that for each A A 0, B>B 0 /p,andallf S e Rn one has 2 js ζ j,a,b f 2 js ζ j,a,b f 1/Lp Rn 1/ L p R n C f F s, p Rn, 11 C f B s, p Rn 12 Since φ j,a,b φ j f for any φ 0, one immediately gets a conseuence of Theorem 2 Corollary 3 The spaces F s, p Rn and F s, p Rn with s R, 0<<,andp P 0 R n with 0<p 0 < min{p,}such that p /p 0 B loc R n areindependentoftheparticular choice of the function φ 0 in Definition 1 The uasinorms arising for different φ 0 are euivalent The proof of Theorem 2 will be given in Section 2 In Section 3 we study the completeness and the lifting property of these spaces by using Theorem 2Wewillusethenotation a bif there exists a constant C>0such that a CbIf a band b awe will write a bfinallyweclaimthatc is always a positive constant but it may change from line to line Other notations will be explained when we meet them 2 Proof of Theorem 2 We will use the idea of [25] by Rychkov to prove Theorem 2 First we need some lemmas Lemma 4 see [25, Theorem 16] Let a function φ 0 D have nonzerointegral,andletφx = φ 0 x 2 n φ 0 x/2thenfor

3 Function Spaces 3 any N 0there exist two functions ψ 0, ψ D, suchthatψ has vanishing moments up to order N and f= ψ j φ j f, f D, 13 where ψ j x := 2 jn ψ j 2 j x and φ j x := 2 jn φ2 j x for j N Let R n = I1,whereI 1 is the set of all unit dyadic cubes in R n Thenitiseasytoget K B f x = R n fy 2 B x y dy 19 2 B distx, fy dy I 1 Before the next lemma we denote a special convolution operator which is given by > 1, by Minkowski s ineuality and Hölder s ine- Since p uality, K B f x := R n fy 2 B x y dy B 0 14 Lemma 5 see [25, Lemma210] Let 0<r<, φ 0 D, R n φ 0 dx =0,andφ=φ 0 2 n φ 0 /2 and A>n/r, B 0 Then there is a constant C depending only on n, r, φ 0, A, B such that for all f S e Rn and each x R n, j N 0,onehas φ j,a,b fxr C k=j 2 j kar n {M loc φ k f r x +K Br φ k f r x} 15 Lemma 6 see [24, Corollary 32] Let p B loc R n and 1<<; then there exists a positive constant C such that for all seuences {f j } of locally integrable functions on Rn M loc f j 1/Lp Rn C f j 1/Lp Rn 16 Lemma 7 see [23,Lemma2114] Let X be a real or complex vector space and be a semimodular on XThen x 1and x 1 are euivalent If is continuous, then also x <1 and x < 1 are euivalent, as are x =1and x = 1 Lemma 8 Let p B loc R n and 1<< Then there exists a positive constant B 0 =B 0 n > 0 such that for B B 0 /p K B f i 1/Lp Rn C f i 1/Lp Rn, 17 where C is a positive constant and {f i } are locally integrable functions on R n Proof By homogeneity, it suffices to consider the case f i 1/Lp Rn 1 18 K B f i x 1/ [ 2 B distx, I [ 1 I 1 { { { I 1 2 B distx, [ B distx,px/2 2 [ 2 B distx,p x/2 I 1 f i y dy ] ] f i y dy ] f i y dy ]px/ } } } 1/p x 1/ 1/ 1/px 20 Since B>0and p >1, the latter factor is uniformly bounded in x We take the pxth power of the above ineuality and integrate it We get R n K B f i x px/ dx 2 I 1 R B distx,px/2 dx n [ It is easy to know that for p >1andalsothat f i y dy ] px/ 21 2 R B distx,px/2 dx dx 22 n f i y dy M locf i x, x 23

4 4 Function Spaces By these two observations and 21, we have R n [ R n K B f i x px/ dx M loc f i x ] Applying Lemmas 6 and 7 we obtain R n K B f i x Using Lemma 7 againweobtain Thus we have K B f i 1/Lp Rn px/ K B f i 1/Lp Rn px/ dx 24 dx f i 1/Lp Rn 27 We give a notation of norm in L p l which will be used in the following context: {g j} := L p l g j 1/Lp Rn 28 Lemma 9 see [27, Lemma2] Let 0<, δ>0for any seuence {g j } of nonnegative numbers denote 0 Then G j = 2 k j δ g k 29 {G j} C {g 0 l j} 0 30 l holds, where C is constant and only depends on, δ Lemma 10 see [19, Lemma3] Let 0<, δ>0,and p P 0 R n For any seuence {g j } of nonnegative measurable functions on R n 0,denote Then G j x = 2 k j δ g k x, x R n 31 {G j} C 0 L p l 1 {g j} 0 {G j} C 0 l L p 2 {g j} 0 L p l, l L p 32 hold with some constants C 1 =C 1, δ and C 2 =C 2 p,, δ Proof of Theorem 2 By Lemma 4,takeψ 0, ψ D,withlarge enough L ψ so that 13istrueItfollowsthat where ζ j fx y R n ζ j ψ k z φ k fx y z dz I jk sup z R n φ k fx y z, m k,a,b z 33 I jk := R n ζ j ψ k z m k,a,b z dz 34 Because of the elementary ineuality m j,a,b z m j,a,b y m j,a,b z y, m j,a,b y { m k,a,b y, j k, { 2 j ka m k,a,b y, j < k { we have the following fact: ζ j,a,b 35 I { jk φ k,a,bf x, if j k f x 36 { I jk 2 k ja φ k,a,b f x, if j<k { To estimate I jk,notethat {2k jlζ+1 ζ 2 kn, if j k, j ψ k 2 j klψ+1 2 jn, if j<k, 37 which follows easily from the moment conditions on ζ and ψ Furthermore, ζ j ψ k z is supported in the ball { z max 2 j,2 k },inwhich By the last two estimates, m k,a,b z { 1, if j k 2 k ja, if j<k I jk { 2k jlζ+1, if j k 2 j klψ+1 A, if j<k Weputthisestimatein36 andseethatifwechoosel ψ > 2A [s] andtakeintoaccountl ζ [s], then we arrive at 2 js ζ j,a,b f x 2 ε k j 2 ks φ k,a,bf x 40 with some ε>0itiseasytoseethat,intherightsideof40, wehaveessentiallytheconvolutionwiththeseuence{2 εk },

5 Function Spaces 5 whichisofcourseaboundedoperatoronanyl, 0<< Now by Lemmas 9 and 10 for l and L p R n we easily obtain 2 js ζ j,a,b f 1/Lp Rn 2 ks φ k,a,b f, 1/Lp Rn 2 js ζ j,a,b f 1/ L p R n 2 ks φ j,a,b f L p R n 1/ 41 In other words, we reduce matters to prove 11and12with ζ 0 =φ 0, ζ=φbelowwedoitonlyfor11; the argument for 12issimilar Let 0 < r < and A > A 0 := n/r + max{ s, 0}By Lemma5 and a discrete version of the Hardy ineuality 2 jθ k=j b k τ 2 jθ b k τ θ, τ > 0, 42 Proof We only give the proof for F s, p Rn and for B s, p Rn ;it canbeprovedbythesimilarwayweusethesimilarargument in [25] Let f S e Rn and γ D with supp γ B0, 1We set ζ 0 =γand in the left side of 11 Analyzing the proof of Theorem 2 shows that only finite numbers of derivatives of the kernels are involved in the estimates, and therefore we know sup f γ y y R n 1 + y A 2 y B L p R n f F s, p Rn sup Dα γ, 44 α L where L is a constant and depends on p,, s, n,butnotonf and γ It is easy to know get sup f γx y y R n 1 + y = sup f γu A 2 y B u R n 1+ x u A 2 x u B sup u x 1 f γu 1+ u x A 2 x u B 45 We take L p R n on both sides of the last ineuality and sup f γ y y R n 1 + y A 2 y B L p R n which we apply with θ:=a+s n/rand τ=/r,wehave 2 js φ j,a,b f 1/ = sup f γu u R n 1+ u A 2 u B L p R n sup u 1 f γu 1+ u A 2 u B L p R n 46 2 js {M loc φ j f r x 1/ +K Br φ r /r j f x } 43 f γu 1 sup V 11+ V A 2 χ V B Bu,1 L p R n f γu By 44and46wehave Note that Br p /r B 0 and 1</r<Letr:=p 0 ;by Lemmas 6 and 8,theoperatorsf {M loc f r } 1/r and f {K Br f r } 1/r are all bounded on L p l Hence the desired estimate 11withζ 0 =φ 0, ζ=φ, follows 3 Some Applications In this section, we will consider the completeness, the lifting property,andtherelateduasinormsofthesespacesintroduced in previous section Theorem 11 Let s R, 0<<,andp P 0 R n with 0<p 0 < min{p,}such that p /p 0 B loc R n Thenthe uasinormed spaces B s, p Rn and F s, p Rn are uasi-banach spaces f γx f F s, p Rn sup Dα γ exp N x, 47 α L where N, L are constants and depend on p,, s, n, butnot on f and γ Then we know that the following estimate f γx f F s, p Rn sup { Dα γ x exp N x :x Rn, α L} 48 is valid for all f S e Rn and γ Dwith some constants N, L which may depend on p,, s, n,butnotonf and γthus we obtain that F s, p Rn is continuously embedded in S e Rn Now we conclude the proof of the theorem in a normal way If a seuence of distributions {f j } is Cauchy seuence in

6 6 Function Spaces F s, p Rn,thenby48 it converges pointwise By the completeness of D,theseuencehasalimitf in D Again by 48, we have f S e Rn, since Cauchy seuences are bounded Finally, by Lebesgue s theorem on dominated convergence it is easily seen that f j fin F s, p Rn In next context, we study the action of the Bessel potential operators in our Besov and Triebel-Lizorkin spaces with variable exponent More precisely, we consider the following tdilated version: T a t =id t2 Δ a/2, a R, t > 0, 49 where id denotes the identity operator For f SR n this operator acts by the rule T a t f=ka t f,where K a = F 1 {1 + 4π 2 x 2 a/2 } formally := e 2πi x, 1 + 4π 2 x 2 a/2 dx S R n 50 It is well known that if a>0,thenk a L 1 R n and has the representation K a x u a n e π x 2 /u 2 e u2 /4π du 51 0 u see Stein sbook [28] for these matters, from which it follows rather easily that K a with a>0is C away from the origin and Dα K a x e E x, x 1 52 with an absolute constant E>0 By the identity K a = id Δ N K a+2n, N N, 53 we see that for a 0the distribution K a agrees in R n \{0} with a C function, which again satisfies 52 Bythesameargumentinpage170of[25]weknowthatthe convolution K a t fcan be defined as an element of S e Rn for any f B s, p Rn F s, p Rn,providedthatt t 0 n p The next theorem states explicitly where it acts Theorem 12 the lifting property Let p P 0 R n with 0<p 0 < min{p,}such that p /p 0 B loc R n Then there is a constant t 0 =t 0 n > 0 so that for all 0<<, s R, and every positive t<t 0 p one has T a t :Bs, p Rn B s+a, p R n isomorphically, T a t :Fs, p Rn F s+a, p R n isomorphically 54 Proof The idea of the proof comes from [25] We use again 13withφ and ψ having vanishing moments up to large order L By an argument similar to that one used above to define T a t on Besov and Triebel-Lizorkin spaces with variable exponent, one can establish the identity φ l K a t f= K a t φ l ψ j φ j f, l N 0 From [25]by choosing L sufficiently large, we have 2 ls+a φ l K a t fx 2 ε l j 2 js φ j,a,bf x, ε>0, t< E 2B ln Now by using Theorem 2 with A=A 0 and B=B 0 /p,itfollows easily that if f belongs to B s, p Rn or F s, p Rn,thenK a t f is in B s+a, p R n or F s+a, p R n,respectivelythenthecondition on t becomes t<t 0 p with t 0 =E/2B 0 ln 2 The fact that the maps in 54 are actually onto follows from the identity T a t T a t = id It follows from Theorem 12 that T s t f F 0, p Rn is an euivalent uasinorm on F s, p Rn for small t>0and analogously for B s, p Rn The next theorem gives a version of this result for s N involving pure derivatives Theorem 13 Let s N, 0<<,andp P 0 R n with 0<p 0 < min{p,}such that p /p 0 B loc R n Thenfor any α s f F s, p f B s, p α s Dα f F 0,, f S e, p 57 Dα f B 0,, f S e p α s Proof For brevity, we only give the outline of the proof for Triebel-Lizorkin space with variable exponent; for Besov spacewithvariableexponentitcanbeprovedbysimilarway The ineuality follows immediately from Definition 1 by partial integrationand invokingtheorem 2 The ineualityfors even follows from Theorem 12To obtain it for s odd, it suffices to consider the case s=1inview of Theorem 12,itissufficienttoprovetheestimate K1 t f F 0, Dα f F 0, p p α 1 From [25] again, we have for small t 58 φ j f=2 j n φ ] j f 59 ]=1 x ]

7 Function Spaces 7 All this leads to the following counterpart of 56: φ l K a t fx 2 εl φ 0,A,Bf x + 2 ε l j n j=1 ]=1 φ ] j,a,b f x, x ] ε>0, 60 where, for ] =1,,n, φ ] D has vanishing moments up to order L 1and satisfy φ= n ]=1 x ] φ ] 61 From 58, it is easily deduced by virtue of Theorem 12 Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper Acknowledgments The authors would like to thank the referee for his carefully reading which made the presentation more readable Jingshi Xu is supported by the National Natural Science Foundation of China Grant nos , , and and the Natural Science Foundation of Hainan Province no References [1] P Harjulehto, P Hästö, U V Lê, and M Nuortio, Overview of differential euations with non-standard growth, Nonlinear Analysis: Theory, Methods & Applications, vol72,no12,pp , 2010 [2] M Růžička, Electrorheological Fluids: Modeling and Mathematical Theory,vol1748ofLectureNotes inmathematics,springer, Berlin, Germany, 2000 [3] Y Chen, S Levine, and M Rao, Variable exponent, linear growth functionals in image restoration, SIAM Journal on Applied Mathematics, vol66,no4,p electronic, 2006 [4] P Harjulehto, P Hästö, V Latvala, and O Toivanen, Critical variable exponent functionals in image restoration, Applied Mathematics Letters,vol26,no1,pp56 60,2013 [5] F Li, Z Li, and L Pi, Variable exponent functionals in image restoration, Applied Mathematics and Computation, vol 216, no 3, pp , 2010 [6] O Kováčik and J Rákosník, On spaces L px and W k,px, Czechoslovak Mathematical Journal, vol 41116, no 4, pp , 1991 [7] AAlmeida,JHasanov,andSSamko, Maximalandpotential operators in variable exponent Morrey spaces, Georgian Mathematical Journal,vol15,no2,pp ,2008 [8] A Almeida and P Hästö, Besov spaces with variable smoothness and integrability, Functional Analysis, vol 258, no5,pp ,2010 [9] L Diening, P Hästö, and S Roudenko, Function spaces of variable smoothness and integrability, Functional Analysis,vol256,no6,pp ,2009 [10] B Dong and J Xu, New Herz type Besov and Triebel-Lizorkin spaces with variable exponents, JournalofFunctionSpacesand Applications, vol 2012, Article ID , 27 pages, 2012 [11] J Fu and J Xu, Characterizations of Morrey type Besov and Triebel-Lizorkin spaces with variable exponents, Mathematical Analysis and Applications,vol381,no1,pp , 2011 [12] P Gurka, P Harjulehto, and A Nekvinda, Bessel potential spaces with variable exponent, Mathematical Ineualities & Applications, vol 10, no 3, pp , 2007 [13] P A Hästö, Local-to-global results in variable exponent spaces, Mathematical Research Letters, vol16,no2,pp , 2009 [14] H Kempka, 2-microlocal Besov and Triebel-Lizorkin spaces of variable integrability, Revista Matemática Complutense,vol22, no 1, pp , 2009 [15] H Kempka, Atomic, molecular and wavelet decomposition of generalized 2-microlocal Besov spaces, Function Spaces and Applications,vol8,no2,pp ,2010 [16] E Nakai and Y Sawano, Hardy spaces with variable exponents and generalized Campanato spaces, JournalofFunctionalAnalysis,vol262,no9,pp ,2012 [17] C Shi and J Xu, Herz type Besov and Triebel-Lizorkin spaces with variable exponent, Frontiers of Mathematics in China,vol 8, no 4, pp , 2013 [18] J S Xu, The relation between variable Bessel potential spaces and Triebel-Lizorkin spaces, Integral Transforms and Special Functions,vol19,no7-8,pp ,2008 [19] J Xu, Variable Besov and Triebel-Lizorkin spaces, Annales Academiæ Scientiarum Fennicæ Mathematica,vol33,no2,pp , 2008 [20] J Xu, An atomic decomposition of variable Besov and Triebel- Lizorkin spaces, Armenian Mathematics,vol2,no1, pp 1 12, 2009 [21] D Cruz-Uribe, A Fiorenza, J M Martell, and C Pérez, The boundedness of classical operators on variable L p spaces, Annales Academiæ Scientiarum Fennicæ Mathematica, vol31, no 1, pp , 2006 [22] D Cruz-Uribe, A Fiorenza, and C J Neugebauer, The maximal function on variable L p spaces, Annales Academiæ Scientiarum Fennicæ Mathematica,vol28,no1,pp ,2003 [23] L Diening, P Harjulehto, P Hästö, and M Růžička, Lebesgue and Sobolev Spaces with Variable Exponents,vol2017ofLecture Notes in Mathematics, Springer, Heidelberg, Germany, 2011 [24] AGogatishvili,ADanelia,andTKopaliani, LocalHardy-Littlewood maximal opeator in variable Lebesgue spaces, Banach Mathematical Analysis,vol8,no2,2014 [25] V S Rychkov, Littlewood-Paley theory and function spaces with A loc p weights, Mathematische Nachrichten, vol224,pp , 2001 [26] T Schott, Function spaces with exponential weights I, Mathematische Nachrichten,vol189,pp ,1998

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