Boundedness of fractional integrals on weighted Herz spaces with variable exponent

Izuki and Noi Journal of Inequalities and Applications 206) 206:99 DOI 0.86/s3660-06-42-9 R E S E A R C H Open Access Boundedness of fractional integrals on weighted Herz spaces with variable exponent Mitsuo Izuki * and Takahiro Noi 2 * Correspondence: izuki@okayama-u.ac.jp Faculty of Education, Okayama University, 3-- Tsushima-naka, Okayama, 700-8530, Japan Full list of author information is available at the end of the article Abstract Our aim is to prove the boundedness of fractional integral operators on weighted Herz spaces with variable exponent. Our method is based on the theory on Banach function spaces and the Muckenhoupt theory with variable exponent. MSC: 42B35 Keywords: Herz spaces; Muckenhoupt weight; variable exponent Introduction The boundedness of fractional integrals on function spaces is one of the important problems not in harmonic analysis but also in potential theory and in partial differential equations. Among the development of variable exponent analysis the we can list up boundedness of fractional integrals on function spaces with variable exponent. Capone, Cruz- Uribe and Fiorenza [] have proved the boundedness on ebesgue spaces with variable exponent, provided that the exponents satisfy the log-hölder continuous conditions. The conditions on variable exponents have been established by the study of the boundedness of the Hardy-ittlewood maximal operator on spaces with variable exponent [2 8]. Cruz- Uribe, Fiorenza, Martell and Pérez [9] have also proved the boundedness of fractional integrals by virtue of the extrapolation method. Every Herz space has an interesting norm involving both local and global information and has been studied in harmonic analysis. u and Yang [0] have proved the boundedness of fractional integrals on Herz spaces. Inspired by the study of variable exponent analysis and on Herz spaces the first author has defined Herz spaces with variable exponent [, 2]. ater he has proved the boundedness of fractional integrals on Herz spaces with variable exponent [3]. Almeida and Drihem [4] have also independently provedthe boundedness. Based on the Muckenhoupt theory [5] the modern harmonic analysis has been greatly developed. Recently the generalized Muckenhoupt weights with variable exponent have been considered [6 20]. In particular, Cruz-Uribe, Fiorenza and Neugebauer [7] and Diening and Hästö [8] have independently proved the equivalence between the Muckenhoupt condition and the boundedness of the Hardy-ittlewood maximal operator on weighted ebesgue spaces in the variable exponent setting. We also note that Cruz- Uribe and Wang [2] have obtained the boundedness of fractional integrals on weighted ebesgue spaces with variable exponent applying the extrapolation theorem. 206 Izuki and Noi. This article is distributed under the terms of the Creative Commons Attribution 4.0 International icense http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

IzukiandNoi Journal of Inequalities and Applications 206) 206:99 Page 2 of 5 In this paper we define weighted Herz spaces with variable exponent and prove the boundedness of fractional integrals on those spaces under proper assumptions on weights and exponents. Our argument is based on the theory on Banach function spaces and on the Muckenhoupt theory with variable exponent. The authors have also considered other problems on boundedness of some operators on weighted Herz spaces with variable exponent in the recent preprints [22, 23]. In this paper we use the following symbols and notation:. For any measurable set E, E denotes the ebesgue measure and χ E means the characteristic function. 2. A locally integrable and positive function defined on R n is said to be a weight. We write we):= E wx)dx for a weight w and a measurable set E. 3. If there existsa positive constant C independent of the main parameters such that A CB,thenwewriteAB. 2 Preliminaries 2. Variable ebesgue spaces Based on the fundamental papers and books [2, 7, 8, 24] we introduce ebesgue spaces with variable exponent. et p ):R n [, ) be a measurable function. The variable exponent ebesgue space p ) R n ) is the set of all complex-valued measurable functions f defined on R n satisfying ρ p f ):= f x) px) dx <. R n It is well known that the variable exponent ebesgue space p ) R n ) becomes a Banach space equipped with a norm given by { ) } f f p ) := inf λ >0:ρ p. λ Denote by PR n ) the set of all measurable functions p ):R n, )suchthat <p := ess inf px), p+ := ess sup px)<. ) x R n x R n Ameasurablefunctionp ) definedonr n is said to be globally log-hölder continuous if it satisfies px) py) log x y ) x, y R n, x y /2 ), 2) px) p loge + x ) x R n ) 3) for some real number p.thesetofp )satisfying2)and3)isdenotedbyhr n ). It is also well known that the Hardy-ittlewood maximal operator M,definedby Mf x):= sup B:ball,x B f y) dy B B is bounded on p ) R n )wheneverp ) PR n ) HR n )[3 6].

IzukiandNoi Journal of Inequalities and Applications 206) 206:99 Page 3 of 5 2.2 The Muckenhoupt weights with variable exponent et p ) PR n )andw be a weight. The weighted variable exponent ebesgue space p ) w) is the set of all complex-valued measurable functions f such that fw /p ) p ) R n ). The space p ) w) is a Banach space equipped with the norm f p ) w) := fw /p ) p ). Below p ) istheconjugateexponentofp ) givenby/p )+/p )=.Nowwedefine the Muckenhoupt classes. We begin with the classical Muckenhoupt A weight. Definition AweightissaidtobeaMuckenhouptA weight if Mwx) wx) holdsfor almost every x R n.theseta consists of all Muckenhoupt A weights. The original Muckenhoupt A p class with constant exponent p, ) establishedby Muckenhoupt [5] canbe generalized in terms of a variable exponent asfollows. Definition 2 Suppose p ) PR n ). A weight w is said to be an A p ) weight if sup B:ball w /p ) χ B p ) w /p ) χ B p B ) <. Our symbol A p ) slightly differs from that in [7, 8, 2]wherethespace p ) w)isdefined as the set consisting of all f such that fw p ) R n ). If p ) equalsaconstantp, ) in Definition 2, then we see immediately that the definition is equivalent to the classical Muckenhoupt class A p [5]. Diening and Hästö [8] have pointed out that Definition 2 does not directly imply the monotone property of the class A p ). In order to obtain the property they have generalized themuckenhouptclassasfollows. Definition 3 Suppose p ) PR n ). A weight w is said to be an à p ) weight if sup B:ball B p B wχ B w χ B p )/p ) <, where p B := B B px) dx) is the harmonic average of p )overb.thesetã p ) consists of all à p ) weights. Based on the definition à p ) Diening and Hästö [8], emma 3., have proved the next monotone property. Theorem Suppose p ), q ) PR n ) HR n ) and p ) q ). Then we have A à p ) à q ). We next state the relation between the generalized Muckenhoupt conditions and the boundedness of the Hardy-ittlewood maximal operator on weighted ebesgue spaces in the variable exponent setting. Theorem 2 Suppose p ) PR n ) HR n ). Then the following three conditions are equivalent:

IzukiandNoi Journal of Inequalities and Applications 206) 206:99 Page 4 of 5 A) w A p ). B) w à p ). C) The Hardy-ittlewood maximal operator is bounded on the weighted variable ebesgue space p ) w). Cruz-Uribe, Fiorenza and Neugebauer [7]haveprovedA) C). On the other hand, Diening and Hästö [8] haveprovedb) C). By Theorem 2 we can identify A p ) and à p ), provided that p ) PR n ) HR n ). Combining Theorems and 2 we get the monotone property for the class A p ), that is, the next corollary is true. Corollary If p ), q ) PR n ) HR n ) and p ) q ), then we have A A p ) A q ). In order to state the boundedness of fractional integrals on weighted function spaces we shall define the class Ap ), p 2 )) as follows. Definition 4 et 0 < β < n and p ), p 2 ) PR n )suchthat/p 2 x) /p x) β/n. Aweightw is said to be an Ap ), p 2 )) weight if wχ B p 2 ) w χ B p ) B β n holds for all balls B R n. emma et 0<β < nandp ), p 2 ) PR n ) such that /p 2 x) /p x) β/n. Then w Ap ), p 2 )) if and only if w p2 ) A +p2 )/p ). Proof Cruz-Uribe and Wang [2], Proposition 5.4, have proved a result similar to the lemma. KomoriandMatsuoka [25] have also considered the case with constant exponent. We will prove the lemma by referring to [2, 25]. Note that p 2 ) = + p 2 ) p ) p 2 ) + β n p ) = by the assumption on p )andp 2 ). Below we fix a ball B R n arbitrarily. It is easy to see that holds. w p 2 ) )+ p 2 ) p ) ) χ B B + p 2 ) p ) w p 2 ) ) + = p 2 )+ p 2 ) w p ) ) p2 )+ χ B B + p 2 w ) p ) = B wχ β/n B w χ p 2 ) B β/n p 2 ) p ) ) χ B p 2 ) p ) ) χ B + p 2 ) p ) ) + p 2 ) p ) ) p ) 4)

IzukiandNoi Journal of Inequalities and Applications 206) 206:99 Page 5 of 5 If w Ap ), p 2 )), then we have w p 2 ) )+ p 2 ) p ) ) χ B B + p 2 ) p ) w p 2 ) ) + = B wχ β/n B w χ p 2 ) B β/n. p ) This implies that w p 2 ) A + p 2 ) p ). If w p 2 ) A + p 2 ) p ),then,by4), we see that wχ B p 2 ) w χ B p ) p 2 ) p ) ) χ B + p 2 ) p ) ) = w p 2 ) )+ p 2 ) p ) ) χ B β/n + p 2 w p 2 ) ) + p 2 ) ) p ) ) χ B β/n p ) p B β/n + p 2 ) ) ) holds. Hence we have w Ap ), p 2 )). This completes the proof. 2.3 Herz spaces with variable exponent et R n be a measurable set and w a positive and locally integrable function on.the set p ) loc, w) consists of all functions f satisfying the following condition: for all compact sets E there exists a constant λ >0suchthat f x) λ E px) wx)dx <. et l Z. We use the following notations in order to define Herz spaces: B l := { x R n : x 2 l}, R l := B l \ B l, χ l := χ Rl. Definition 5 et p ) PR n ), 0 < q <, andα R. The homogeneous weighted Herz space K α,q p ) w) is the collection of f p ) loc Rn \{0}, w)suchthat f K α,q p ) w) := /q 2 αkq f χ k w)) p ) <. 5) Herz spaces with variable exponent were initially defined by the first author [, 2]. The weighted case has been recently studied by the authors [22, 23]. 2.4 Weighted Banach function spaces We introduce Banach function space and state fundamental properties of it based on the book [26] by Bennett and Sharpley. We additionally show some properties of Banach function spaces in terms of the boundedness of the Hardy-ittlewood maximal operator. We will also consider the weighted case based on [27] by Karlovich and Spitkovsky.

IzukiandNoi Journal of Inequalities and Applications 206) 206:99 Page 6 of 5 Definition 6 et M be the set of all complex-valued measurable functions defined on R n,andx alinearsubspaceofm.. The space X is said to be a Banach function space if there exists a functional X : M [0, ] satisfying the following properties: et f, g, f j M j =,2,...), then a) f X holds if and only if f X <. b) Norm property: i. Positivity: f X 0. ii. Strict positivity: f X =0holds if and only if f x)=0for almost every x R n. iii. Homogeneity: λf X = λ f X holds for all λ C. iv. Triangle inequality: f + g X f X + g X. c) Symmetry: f X = f X. d) attice property: If 0 gx) f x) for almost every x R n,then g X f X. e) Fatou property: If 0 f j x) f j+ x) for all j and f j x) f x) as j for almost every x R n,thenlim j f j X = f X. f) For every measurable set F R n such that F <, χ F X is finite. Additionally there exists a constant C F >0depending only on F so that F hx) dx C F h X holds for all h X. 2. Suppose that X is a Banach function space equipped with a norm X.The associated space X is defined by X := { f M : f X < }, where { } f X := sup f x)gx)dx : g X. g R n The proof of the following fundamental lemma is found in [26]. emma 2 et X be a Banach function space. Then the following hold:. The associated space X is also a Banach function space. 2. The orentz-uxemburg theorem.) X ) = X holds, in particular, the norms X ) and X are equivalent. 3. The generalized Hölder inequality.) If f X and g X, then we have f x)gx) dx f X g X. R n An easy application of the generalized Hölder inequality gives us the following lemma. emma 3 If X is a Banach function space, then we have that for all balls B, B χ B X χ B X.

IzukiandNoi Journal of Inequalities and Applications 206) 206:99 Page 7 of 5 Kováčik and Rákosník [24] have proved that the generalized ebesgue space p ) R n ) with variable exponent p ) is a Banach function space and the associate space is p ) R n ) with norm equivalence. If we assume some conditions for the boundedness of the Hardy-ittlewood maximal operator M on X, then the norm X has properties similar to the classical Muckenhoupt weights. emma 4 et X be a Banach function space. Suppose that the Hardy-ittlewood maximal operator M is weakly bounded on X, that is, χ {Mf >λ} X λ f X 6) is true for all f Xandallλ >0.Then we have sup B:ball B χ B X χ B X <. 7) The proof of emma 4 is found in the first author s paper [9], emmas 2.4 and 2.5, and [28], emmas G and H. Remark If M is bounded on X, thatis, Mf X f X holds for all f X, thenwecan easily check that 6) holds. On the other hand, if M is bounded on the associate space X, then emma 2 shows that 7)istrue. Below we define weighted Banach function space and give some of its property. et X be a Banach function space. The set X loc R n ) consists of all measurable functions f such that f χ E X for any compact set E with E <. Given a function W such that 0 < Wx)< for almost every x R n, W X loc R n )andw X ) loc R n ), we define the weighted Banach function space X R n, W ) := {f M : fw X}. Then the following hold. emma 5. The weighted Banach function space XR n, W) is a Banach function space equipped the norm f XR n,w) := fw X. 2. TheassociatespaceofXR n, W) is a Banach function space and equals X R n, W ). The properties above naturally arise from those of the usual Banach function spaces and the proof is found in [27]. Remark 2 et p ) PR n ). Comparing the definition of XR n, W) with weighted ebesgue spaces p ) w p ) )and p ) w p ) ) respectively, we obtain the following:. If we take X = p ) R n ) and W = w,thenwehave p ) R n, w)= p ) w p ) ).

IzukiandNoi Journal of Inequalities and Applications 206) 206:99 Page 8 of 5 2. If we take X = p ) R n ) and W = w,thenwehave p ) R n, w )= p ) w p ) ). Therefore emma 5 yields p ) w p ))) = p ) R n, w )) = p ) R n, w ) = p ) w p ) ). The next lemma has been initially proved by the first author [2] inthecasethatx = p ) R n ), however, his argument depends on Diening s work [6]. Cruz-Uribe, Hernández and Martell [29] have recently given its alternative proof based on the Rubio de Francia algorithm [30 32]. As is mentioned in the authors preprint [22], the proof due to [29]is self-contained and valid for general Banach function spaces X. emma 6 et X be a Banach function space. Suppose that M is bounded on the associate space X. Then there exists a constant 0<δ <such that for all balls B R n and all measurable sets E B, χ E X χ B X ) E δ. B Proof For the reader s convenience we shall give the proof based on [29]. et A := and define a function M X X where Rgx):= k=0 M k gx) 2A) k g X ), 8) g k =0), M k g := Mg k =), MM k g) k 2). For every g X, the function Rg satisfies the following properties:. gx) Rgx) for almost every x R n. 2. Rg X 2 g X, namely the operator R is bounded on X. 3. MRg)x) 2ARgx),thatis,Rg is a Muckenhoupt A weight.wenotethatthe constant 2A appearing in the right-hand side is independent of g and x. We can write RgS)= S Rgx)dx for every measurable set S Rn because Rg is a weight. Thus by virtue of [33], Chapter 7, we can take positive constants C and δ < depending only on A and n so that, for all balls B and all measurable sets E B, ) RgE) E δ RgB) C B holds. Now we fix g X with g X arbitrarily. By virtue of generalized Hölder s inequality we have χe x)gx) ) E δ ) E δ dx RgE) C RgB) C χ B X Rg X R n B B ) E δ χ B X. B

IzukiandNoi Journal of Inequalities and Applications 206) 206:99 Page 9 of 5 Therefore by the duality we get { } χ E X sup χ E x)gx)dx : g X, g X g R n ) E δ χ B X. B This completes the proof of the lemma. 3 Themainresults 3. Remarks on boundedness of fractional integrals on ebesgue spaces Definition 7 Given p ) PR n )andaweightw,wesayp ), w)isanm-pair if the maximal operator M is bounded on p ) w p ) )andon p ) w p ) ). et 0 < β < n. Then the fractional integral operator I β is defined by I β f y) f x):= dy. R n x y n β Cruz-Uribe and Wang [2], Corollary 3.7, have obtained the following boundedness of I β on weighted variable exponent ebesgue spaces. Theorem 3 et p ) PR n ) HR n ), 0 < β < n/p + and σ := n/β). Define p 2 ) by /p 2 ) /p ) β/n. Then for all weights w such that p 2 )/σ, w σ ) is an M-pair, I β is bounded from p ) w p ) ) to p2 ) w p2 ) ). et w Ap ), p 2 )). Note that p 2 )/σ =+p 2 )/p ). emma and the equivalence w p 2 ) A +p2 )/p ) = A p2 )/σ w σ p 2 )/σ ) A p2 )/σ ) imply that M is bounded on p2 )/σ w σ p2 )/σ )andon p 2 )/σ ) w σ p 2 )/σ ) ). Therefore, p 2 )/σ, w σ )isanm-pair whenever w A p ), p 2 ) ), p ), p 2 ) P R n) H R n) and 0 < β < n/p +. Hence we have the following corollary. Corollary 2 et p ) PR n ) HR n ) and 0<β < n/p +. Define p 2 ) by /p 2 ) /p ) β/n. If w Ap ), p 2 )), then I β is bounded from p ) w p ) ) to p2 ) w p2 ) ). 3.2 Boundedness of fractional integrals on Herz spaces et p 2 ) PR n ) HR n )andw p2 ) A. Then the monotone property yields w p2 ) A p2 ). Hence the Hardy-ittlewood maximal operator M is bounded on p2 ) w p2 ) ). On the other hand, by the definition of A p ) it is easy to see that w p2 ) A p2 ) implies w p 2 ) A p 2 ). ThusM is bounded on p 2 ) w p 2 ) ). Therefore applying Remark 2, emma 6, and the orentz-uxemburg theorem we can take constants δ, δ 2 0, ) such that χ E p 2 ) w p 2 ) ) = χ B p 2 ) w p 2 ) ) χ E p 2 ) w p 2 ) )) χ B p 2 ) w p 2 ) )) ) E δ, 9) B

IzukiandNoi Journal of Inequalities and Applications 206) 206:99 Page 0 of 5 χ E p 2 ) w p 2 ) )) χ B p 2 ) w p 2 ) )) ) E δ2 0) B for all balls B and all measurable sets E B. Now we take a positive number β so that 0<β < nδ + δ 2 ). Then we can take a real number α such that nδ < α < nδ 2 β because the choice of β shows nδ < nδ 2 β. Usingp 2 ) andβ, we additionally define p ) so that /p 2 ) /p ) β/n holds. We see that p ) satisfiesp ) PR n ) HR n )and 0<β < n/p +. Applying the monotone property again we have wp2 ) A A +p2 )/p ). Thus by emma we get w Ap ), p 2 )). Therefore in the setting as above we can apply Corollary 2 to obtain the boundedness of the fractional integral operator I β on Herz spaces. Theorem 4 et 0<q q 2 <, p 2 ) PR n ) HR n ), w p2 ) A, δ, δ 2 0, ) be the constants appearing in 9) and 0) respectively, 0<β < nδ + δ 2 ) and nδ < α < nδ 2 β. Define p ) by /p 2 ) /p ) β/n. Then the fractional integral operator I β is a bounded operator from K α,q 2 p 2 ) wp2 ) ) to K α,q p ) wp ) ). Proof et f K α,q p ) wp ) ). Then, by the Jensen inequality, we have I β f q K α,q = 2 2 αq 2k I β f ) ) q q 2 χ p 2 ) wp 2 ) ) k q 2 p 2 ) w p 2 ) ) 2 αq k I β f ) χ k q p 2 ) w p 2 ). ) ) et f j := f χ j for any j Z.Thenf = j= f j.sowehave I β f q K α,q 2 p 2 ) wp 2 ) ) + + 2 αq k I β f ) χ k q p 2 ) w p 2 ) ) 2 αq k 2 αq k 2 αq k I β ) f j χk p 2 ) w p 2 ) ) j k 2 k+ I β ) f j χk p 2 ) w p 2 ) ) j=k I β ) f j χk p 2 ) w p 2 ) ) =: U + U 2 + U 3. 2) Step. We estimate U. By the definition of I β and the generalized Hölder inequality, we obtain I β ) f j x) χk x) χ k x) x y β n fj y) dy R n 2 kβ n) f j p ) w p ) ) χ j p ) w p ) )) χ kx).

IzukiandNoi Journal of Inequalities and Applications 206) 206:99 Page of 5 By taking the p 2 ) w p 2 ) )-norm and using emma 4,wehave I β f j ) χk p 2 ) w p 2 ) ) 2 kβ n) f j p ) w p ) ) χ j p ) w p ) )) χ B k p 2 ) w p 2 ) ) =2 kβ f j p ) w p ) ) χ j p ) w p ) )) 2 kn χ Bk p 2 ) w p 2 ) ) 2 kβ f j p ) w p ) ) χ j p ) w p ) )) χ B k p 2 ) w p 2 ) )). By 0), we see that I β f j ) χk p 2 ) w p 2 ) ) 2 kβ f j p ) w p ) ) χ j p ) w p ) )) χ B k p 2 ) w p 2 ) )) =2 kβ f j p ) w p ) ) χ j p ) w p ) )) χ B j p 2 ) w p 2 ) )) χ Bj p 2 ) w p 2 ) )) χ Bk p 2 ) w p 2 ) )) 2 kβ 2 nδ 2j k) f j p ) w p ) ) χ j p ) w p ) )) χ B j p 2 ) w p 2 ) )). 3) By the obvious inequality 2 jβ χ Bj x) I β f Bj )x) and the boundedness of I β : p ) w p ) ) p2 ) w p2 ) ), we have χ Bj p 2 ) w p 2 ) ) 2 jβ I β χ Bj p 2 ) w p 2 ) ) 2 jβ χ Bj p ) w p ) ). By using emma 4 again, we obtain χ Bj p 2 ) w p 2 ) ) 2 jβ χ Bj p ) w p ) ) 2jn β) χ Bj p ) w p ) )) 2 jn β) χ j p ) w p ) )). 4) Combining 3) and4) and using emma 3, we get I β f j ) χk p 2 ) w p 2 ) ) 2 kβ 2 nδ2j k) 2 jn β) f j p ) w p ) ) χ B j p 2 ) w p 2 ) χ ) B j p 2 ) w p 2 ) )) =2 β nδ2)k j) f j p ) w p ) ) 2 jn χ Bj p 2 ) w p 2 ) ) χ ) B j p 2 ) w p 2 ) )) 2 β nδ 2)k j) f j p ) w p ) ). Thus we get U j k 2 2αj 2 β nδ2+α)k j) f j p ) w p ) ) )q.notethatβ nδ 2 + α <0.Weconsiderthetwocases <q < and 0<q. If < q <, then, by using the Hölder inequality, we obtain U 2 αj 2 β nδ2+α)k j) f j p ) w p ) ) j k 2 j k 2 ) 2 αjq 2 β nδ 2+α)k j)q /2 f j q 2 β nδ 2+α)k j)q /q p ) w p ) /2 ) j k 2

IzukiandNoi Journal of Inequalities and Applications 206) 206:99 Page 2 of 5 = j k 2 2 αjq f j q j= f q K α,q p ) wp ) ). 2 αjq 2 β nδ 2+α)k j)q /2 f j q p ) w p ) ) 2 β nδ 2+α)k j)q /2 p ) w p ) ) k j 2 If 0 < q, then by using the Jensen inequality we obtain U = 2 αj 2 β nδ2+α)k j) f j p ) w p ) ) j k 2 j k 2 2 αjq f j q j= f q K α,q p ) wp ) ). 2 αjq 2 β nδ 2+α)k j)q f j q p ) w p ) ) 2 β nδ 2+α)k j)q p ) w p ) ) k j 2 Step 2. We estimate U 2.UsingCorollary2 and k j, it is easy to see that U 2 = = 2 αq k k+ j=k k+ I β ) f j χk p 2 ) w p 2 ) ) j=k f q K α,q p ) wp ) ). 2 αk j) 2 αj f j p ) w p ) ) Step 3. We estimate U 3. Using the generalized Hölder inequality, we have, for every j, k Z with j k +2, I β f j )χ k x) 2 j n+β) f y) dy χk x) B j 2 j n+β) fw p ) χ j p ) w p ) ) χ kx). By taking the p 2 ) w p 2 ) )-norm and 9), we have I β f j ) χk p 2 ) w p 2 ) ) 5) 2 j n+β) f j w p ) χ j p ) w p ) χ k ) p 2 ) w p 2 ) ) 2 j n+β) f j w p ) χ j p ) w p ) χ χ k p 2 j ) w p 2 ) ) ) p 2 ) w p 2 ) ) χ j p 2 ) w p 2 ) ) 2 j n+β) 2 nδk j) f j w p ) χ j p ) w p ) χ j ) p 2 ) w p 2 ) ). 6)

IzukiandNoi Journal of Inequalities and Applications 206) 206:99 Page 3 of 5 By the definition of Ap ), p 2 )), we obtain χ j p ) w p ) ) χ j p 2 ) w p 2 ) ) χ B j p ) w p ) ) χ B j p 2 ) w p 2 ) ) w χ Bj p ) wχ Bj p 2 ) 2 jn β/n). Hence we have I β f j ) χk p 2 ) w p 2 ) ) 2 j n+β) 2 nδ k j) f j w p ) χ j p ) w p ) ) χ j p 2 ) w p 2 ) ) 2 j n+β) 2 nδ k j) 2 jn β/n) f j w p ) =2 nδ k j) f j w p ). Therefore we see that U 3 = 2 αq k I β ) f j χk p 2 ) w p 2 ) ) 2 αq k 2 nδ k j) f j w p ) q 2 α+nδ)k j) 2 αj f j w p )). Note that α + nδ >0.Weconsiderthetwocases: <q < and 0<q. If < q <, then by using the Hölder inequality we obtain U 3 2 α+nδ)k j) 2 αj f j w p ) 2 αqj f j w q j= f q K α,q p ) wp ) ). 2 α+nδ )k j)q /2 2 αq j f j w q p ) ) 2 α+nδ )k j)q /2 p ) k j 2 If 0 < q, then by using the Jensen inequality we obtain U 3 2 α+nδ)k j) 2 αj f j w p ) 2 α+nδ )k j)q 2 αq j f j w q p ) 2 α+nδ )k j)q /2 /q

IzukiandNoi Journal of Inequalities and Applications 206) 206:99 Page 4 of 5 2 αqj f j w q j= f q K α,q p ) wp ) ). 2 α+nδ )k j)q p ) k j 2 Consequently we have proved the theorem. 3.3 The non-homogeneous case In this paper we have defined the weighted Herz space with variable exponent and proved the boundedness of the fractional integrals on the spaces in the homogeneous case. Our argument is also valid for the non-homogeneous case. Definition 8 et p ) PR n ), 0 < q <,andα R. For non-negative integer k,let R k k ), C k := B 0 k =0). The non-homogeneous weighted Herz space K α,q p ) w) is the collection of f p ) loc Rn, w) such that f α,q K p ) w) := /q 2 αkq f χ Ck w)) p ) <. k=0 Theorem 5 et 0<q q 2 <, p 2 ) PR n ) HR n ), w p2 ) A, δ, δ 2 0, ) be the constants appearing in 9) and 0) respectively,0<β < nδ + δ 2 ), and nδ < α < nδ 2 β. Define p ) by /p 2 ) /p ) β/n. Then the fractional integral operator I β is a bounded operator from K α,q 2 p 2 ) wp2 ) ) to K α,q p ) wp ) ). Remark 3 Inthemaintheoremswehaveassumedthatw p2 ) A to obtain the boundedness of I β. One may think that the condition w p2 ) A is too strong. In this paper we could not prove the main theorems under weaker conditions on w. Competing interests The authors declare that they have no competing interests. Authors contributions The authors contributed equally to this work. All authors read and approved the final manuscript. Author details Faculty of Education, Okayama University, 3-- Tsushima-naka, Okayama, 700-8530, Japan. 2 Department of Mathematics and Information Science, Tokyo Metropolitan University, Hachioji, 92-0397, Japan. Acknowledgements The first author was partially supported by Grand-in-Aid for Scientific Research C), No. 5K04928, for Japan Society for the Promotion of Science. The second author was partially supported by Grand-in-Aid for Scientific Research C), No. 6K0522, for Japan Society for the Promotion of Science. The authors are thankful to the referees, whose comments have improved the manuscript. Received: 3 April 206 Accepted: 5 August 206 References. Capone, C, Cruz-Uribe, D, SFO, Fiorenza, A: The fractional maximal operator and fractional integrals on variable p spaces. Rev. Mat. Iberoam. 233),743-770 2007)

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