BOUNDEDNESS FOR FRACTIONAL HARDY-TYPE OPERATOR ON HERZ-MORREY SPACES WITH VARIABLE EXPONENT

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1 Bull. Korean Math. Soc , No. 2, pp BOUNDEDNESS FOR FRACTIONA HARDY-TYPE OPERATOR ON HERZ-MORREY SPACES WITH VARIABE EXPONENT Jianglong Wu Abstract. In this paper, the fractional Hardy-type operator of variable order βx is shown to be bounded from the Herz-Morrey spaces M p,q Rn with variable exponent q x into the weighted space M p 2, Rn,ω, where ω = + x γx with some γx > 0 and /q x / x = βx/n when q x is not necessarily constant at infinity. It is assumed that the exponent q x satisfies the logarithmic continuity condition both locally and at infinity that < q q x q + < x R n.. Introduction et f be a locally integrable function on R n. The n-dimensional Hardy operator is defined by H fx := x n ftdt, x R n \{0}. t < x In 995, Christ and Grafakos [2] obtained the result for the boundedness of H on p R n < p < spaces, and they also found the exact operator norms of H on this space. In 2007, Fu et al. [3] gave the central BMO estimates for commutators of n-dimensional fractional and Hardy operators. And recently, author [4, 5, 6, 7, 8, 9] also considers the boundedness for Hardy operator and its commutator in variable exponent Herz-Morrey spaces. The theory of variable exponent ebesgue spaces is started by Orlicz see [0], 93 and Nakano see [, 2], 950 and 95. In particular, the definition of Musielak-Orlicz spaces is clearly stated in []. However, the variable exponent function space, due to the failure of translation invariance and related properties, is very difficult to analyze. Received October 8, 202; Revised January 9, Mathematics Subject Classification. Primary 42B20; Secondary 47B38. Key words and phrases. Herz-Morrey space, Hardy operator, Riesz potential, variable exponent, weighted estimate. 423 c 204 Korean Mathematical Society

2 424 JIANGONG WU Nowadays there is an evident increase of investigations related to both the theory of the spaces q Ω themselves and the operator theory in these spaces. This is caused by possible applications to models with non-standard local growth in elasticity theory, fluid mechanics, differential equations, see for example [3], [4] and references therein and is based on recent breakthrough result on boundedness of the Hardy-ittlewood maximal operator in thesespaces. Byvirtueofthefineworks[5,6,7,8,20,2,22,25,26,27,28], some important conditions on variable exponent, for example, the log-hölder conditions and the Muckenhoupt type condition, have been obtained. Now, we define the n-dimensional fractional Hardy-type operators of variable order βx as follows. Definition.. et f be a locally integrable function on R n, 0 βx < n. The n-dimensional fractional Hardy-type operators of variable order βx are defined by.0a.0b where x R n \{0}. H β fx := H β fx := ftdt, x n βx t < x t x ft t n βxdt, Obviously, when βx = 0, H β is just H, and denote by H := Hβ = H0. And when βx is constant, H β and Hβ will become H β and Hβ [3], respectively. The Riesz-type potential operator of variable order βx is defined by fy. I β fx = x y n βxdy, 0 < βx < n. R n The boundedness of the operator I β from the space p R n with the variable exponent px into the space q R n with the limiting Sobolev exponent qx = px βx n was an open problem for a long time. It was solved in the case of bounded domains. First, in [29], in the case of bounded domains Ω, there was proved a conditional result: the Sobolev theorem is valid for the potential operator I β within the framework of the spaces p Ω with the variable exponent px satisfying the logarithmic Dini condition, if the maximal operator is bounded in the space p Ω. In 2004, Diening [20] proved the boundedness of the maximal operator. In 2004, Diening [9] proved Sobolev s theorem for the potential I β on the whole space R n assuming that px is constant at infinity px is always constant outside some large ball and satisfies the same logarithmic condition as in [29]. Another progress for unbounded domains is the result of Cruz-Uribe

3 BOUNDEDNESS FOR FRACTIONA HARDY-TYPE OPERATOR 425 et al. [8] on the boundedness of the maximal operator in unbounded domains for exponents px satisfying the logarithmic smoothness condition both locally and at infinity. In [24], Kokilashvili and Samko prove Sobolev-type theorem for the potential I β from the space p R n into the weighted space ω q R n with the power weight ω fixed to infinity, under the logarithmic condition for px satisfied locally and at infinity, not supposing that px is constant at infinity but assuming that px takes its minimal value at infinity. Motivated by the above results, we are to investigate mapping properties of the fractional Hardy-type operators H β and Hβ within the framework of the Herz-Morrey spaces with variable exponent. Throughout this paper, we will denote by S the ebesgue measure and by χ S the characteristic function for a measurable set S R n ; Bx,r is the ball centered at x and of radius r; B 0 = B0,. C denotes a constant that is independent of the main parameters involved but whose value may differ from line to line. For any index < qx <, we denote by q x its conjugate index, namely, q x = qx qx C > 0 such thatc D A D.. For A D, we mean that there is a constant 2. Preliminaries In this section, we give the definition of ebesgue and Herz-Morrey spaces with variable exponent, and give basic properties and useful lemmas. 2.. Function spaces with variable exponent et Ω be a measurable set in R n with Ω > 0. We first define ebesgue spaces with variable exponent. Definition 2.. et q : Ω [, be a measurable function. I The ebesgue spaces with variable exponent q Ω is defined by q Ω = {f is measurable function : F q f/η < for some constant η > 0}, where F q f := Ω fx qx dx. The ebesgue space q Ω is a Banach function space with respect to the norm { fx qxdx } f q Ω = inf η > 0 : F q f/η =. η II The space q loc Ω is defined by q loc Ω = {f is measurable : f q Ω 0 for all compact subsets Ω 0 Ω}. III The weighted ebesgue space q ω Ω is defined by as the set of all measurable functions for which f q ω Ω = ωf q Ω <. Ω

4 426 JIANGONG WU Next we define some classes of variable exponent functions. Given a function f loc Rn, the Hardy-ittlewood maximal operator M is defined by Mfx = supr n fy dy, r>0 Bx,r where Bx,r = {y R n : x y < r}. Definition 2.2. Given a measurable function q defined on R n, we write q := essinf x R n qx, q + := esssup x R n qx. I q = essinf x R n q x = q+ q +, q + = esssup x R n q x = q q. II Denote by PR n the set of all measurable functions q : R n, such that < q qx q + <, x R n. III The set BR n consists of all measurable functions q PR n satisfying that the Hardy-ittlewood maximal operator M is bounded on q R n. IV The set C log 0 R n consists of all locally log-hölder continuous functions q : R n 0, satisfies the condition 2. qx qy C ln x y, x y /2, x,y Rn. V The set C log R n consists of all log-hölder continuous at infinity functions q : R n 0, satisfies the condition 2.2 qx q C lne+ x, x Rn, where q = lim x qx. VI Denote by C log R n := C log 0 R n C log R n the set of all globally log-hölder continuous functions q : R n 0,. Remark. The C logrn condition is equivalent to the uniform continuity condition C 2.3 qx qy lne+ x, y x, x,y Rn. The C log R n condition was originally defined in this form in [8]. Next we define the Herz-Morrey spaces with variable exponent. et B k = B0,2 k = {x R n : x 2 k },A k = B k \B k and χ k = χ Ak for k Z. Definition 2.3. Suppose that α R, 0 λ <, 0 < p <, q PR n. The Herz-Morrey space with variable exponent M p,q Rn is defined by M p,q Rn = { f q loc Rn \{0} : f M p,q Rn < },

5 BOUNDEDNESS FOR FRACTIONA HARDY-TYPE OPERATOR 427 where f M p,q Rn = sup 2 k0λ k 0 k 0 Z k= Compare the variable Herz-Morrey space M α,p Herz space q Rn, where { α,p q Rn = f q loc Rn \{0} : k= 2 kαp fχ k p q R n p p,q Rn with the variable 2 kαp fχ k p q R n < }, α,0 Obviously, M p,q Rn α,p = q Rn. In 202, Almeida and Drihem [] discuss the boundedness of a wide class of sublinear operators on Herz spaces K α,p q R n α,p and q R n with variable exponent α and q. In this paper, the author only considers Herz-Morrey α,λ spacem p,q Rn withvariableexponentq butfixedα Randp 0,. However,forthecaseoftheexponentα isvariableaswell,whichcanbefound in the furthermore work for the author Auxiliary propositions and lemmas In this part we state some auxiliary propositions and lemmas which will be needed for proving our main theorems. And we only describe partial results we need. Proposition 2.. et q PR n. I If q C log R n, then we have q BR n. II q BR n if and only if q BR n. The first part in Proposition 2. is independently due to Cruz-Uribe et al.[8] and to Nekvinda [27], respectively. The second of Proposition 2. belongs to Diening [2] see Theorem 8. or Theorem.2 in [7]. Remark 2. Since q x q y qx qy q 2, itfollowsatoncethatifq C log R n, thensodoesq i.e., ifthecondition hold, then M is bounded on q R n and q R n. Furthermore, Diening has proved general results on Musielak-Orlicz spaces. The order βx of the fractional Hardy-type operators in Definition. is not assumed to be continuous. We assume that it is a measurable function on

6 428 JIANGONG WU R n satisfying the following assumptions 2.4 β 0 := essinfβx > 0, x Rn ess suppxβx < n, x R n ess sup x R n p βx < n. In order to prove our main results, we need the Sobolev type theorem for the space R n which was proved in ref. [24] for the exponents px not necessarily constant in a neigbourhood of infinity, but with some extra power weight fixed to infinity and under the assumption that px takes its minimal value at infinity. Proposition 2.2. Suppose that p C log R n PR n. et 2.5 < p px p + <, and βx meet condition 2.4. Then the following weighted Sobolev-type estimate is valid for the operator I β : + x γx I β f f q R n p R, n where qx = px βx n is the Sobolev exponent and 2.6 γx = C βx βx n n 4 C, C being the Dini-ipschitz constant from 2.2 which q is replaced by p. Remark 3. i If βx satisfies the condition of type 2.2: βx β C lne+ x x Rn, then the weight + x γx is equivalent to the weight + x γ. ii One can also treat operator. with βx replaced by βy. In the case of potentials over bounded domains Ω such potentials differ unessentially, if the function βx satisfies the smoothness logarithmic condition as 2., since C x y n βy x y n βx 2 x y n βy in this case see [29], p iii Under the assumptions of Proposition 2.2, similar conclusion is also valid for the fractional maximal operator M β fx = sup r>0 Bx,r n βx Bx,r fy dy.

7 BOUNDEDNESS FOR FRACTIONA HARDY-TYPE OPERATOR 429 iv When p PR n, the assumption that p C log R n is equivalent to assuming /p C log R n, since px py p + 2 px = px py px py. py pxpy p 2 And further, /p C log R n implies that /q C log R n as well. The next lemma known as the generalized Hölder s inequality on ebesgue spaces with variable exponent, and the proof can be found in [5]. emma 2. Generalized Hölder s inequality. Suppose that q PR n, then for any f q R n and any g q R n, we have R n fxgx dx q f q R n g q R n, where C q = +/q /q +. The following lemma can be found in [23]. emma 2.2. et q BR n. I Then there exist positive constants δ 0, and C > 0 such that χ S δ q R n S χ B q R n B for all balls B in R n and all measurable subsets S B. II Then there exists a positive constant C > 0 such that C B χ B q R n χ B q R n for all balls B in R n. Remark 4. i Ifq, C log R n PR n, then we seethat q, q 2 BR n. Hence we can take positive constants 0 < δ < /q +, 0 < δ 2 < / + such that 2.7 χ S q R n χ B q R n δ S, B χ S R n χ B R n δ2 S B hold for all balls B in R n and all measurable subsets S B see [7, 23]. ii On the other hand, Kopaliani [25] has proved the conclusion: If the exponent q PR n equals to a constant outside some large ball, then q BR n if and only if q satisfies the Muckenhoupt type condition sup Q:cube Q χ Q q R χ n Q q R n <.

8 430 JIANGONG WU 3. Main results and their proofs Our main result can be stated as follows. Theorem 3.. Suppose that q C log R n PR n satisfies condition 2.5, and βx meet condition 2.4 which p is replaced by q. Define the variable exponent by x = q x βx n. et 0 < p p 2 <, λ 0, α < λ + nδ, where δ 0,/q + is the constant appearing in 2.7. Then + x γx H β f M p 2, Rn f M p,q Rn, where γx is defined as in 2.6, and C is the Dini-ipschitz constant from 2.2 which q instead of q. Proof. For any f M p,q Rn, if we denote f j := f χ j = f χ Aj for each j Z, then we can write fx = By.0a and emma 2., we have fx χ j x = f j x. H β fx χ k x ft dt χ x n βx k x t < x ft dt χ x n βx k x B k 2 kn x βx 3. ft dt A j 2 kn χ k x f j q Rn χ j q R n x βx χ k x. For Proposition 2.2, we note that I β χ Bk x I β χ Bk x χ Bk x = B k x y n βxdy χ B k x 3.2 C x βx χ Bk x C x βx χ k x.

9 BOUNDEDNESS FOR FRACTIONA HARDY-TYPE OPERATOR 43 Using Proposition 2.2, emma 2.2, 2.7, 3. and 3.2, we have x γx H β f χ k 2 kn 2 kn 2 kn 2 kn χ Bk q R n R n f j χ q R n j + x γx βx χ q R n k f j χ q R n j + x γx I q β χ R n Bk f j q Rn χ j q R n χ B k q Rn χ Bj q R n f j q R n χ Bk q R n 2 j knδ f j q R n. f j q R n χ B j q R n Because of 0 < p /p 2, then we apply inequality p /p a i a i p /p 2, i= and obtain + x γx H β f k0 = sup 2 k0λp k 0 Z k= k0 sup 2 k0λp k 0 Z k= p M 2 kαp 2 2 kαp p 2, Rn i= On the other hand, note the following fact + x γx H β f χ k + x γx H β f χ k R n Rn p p /p 2 2 R n p R n. 3.5 f j q Rn = 2 jα 2 jαp fj p 2 jα j i= q R n /p 2 iαp fi p q R n /p

10 432 JIANGONG WU = 2 jλ α 2 jλ j i= 2 jλ α f M p,q Rn. 2 iαp fi p /p q R n Thus, combining 3.3 and 3.5, and using α < λ+nδ, it follows that + x γx H β f p M p 2, Rn k0 sup 2 k0λp 2 kαp p 2 j knδ f j q k R 0 Z n k= f p M sup k0 2 k0λp 2 kαp 2 j knδ 2 jλ α p p,q Rn k 0 Z k= f p M sup k0 2 k0λp 2 kλp 2 j knδ+λ α p p,q Rn k 0 Z k= f p M sup k0 2 k0λp 2 kλp p,q Rn k 0 Z k= f p M. p,q Rn Consequently, the proof of Theorem 3. is completed. Theorem 3.2. Suppose that q C log R n PR n satisfies condition 2.5, and βx meet condition 2.4 which q instead of p. Define the variable exponent by x = q x βx n. et 0 < p p 2 <, λ 0, α > λ nδ 2, where δ 2 0,/ + is the constant appearing in 2.7. Then + x γx Hβ f M f M p,q Rn p,q Rn, 2 2 where γx is defined as in 2.6, and the Dini-ipschitz constant from condition 2.2 which q is replaced by q. Proof. For simplicity, for any f M fx = p,q Rn, we write fx χ j x = f j x.

11 BOUNDEDNESS FOR FRACTIONA HARDY-TYPE OPERATOR 433 By.0b and emma 2., we have + x γx Hβ fx χ k x ft t x t n βxdt + x γx χ k x ft x βx n dt + x γx χ k x 3.6 R n \B k ft x βx n + x γx dt χ k x A j j=k+ j=k+ Similar to 3.2, we give f j q R + x γx βx n χ n j q χ R n k x. 3.7 I β χ Bj x I β χ Bj x χ Bj x = B j x y n βxdy χ B j x C x βx χ Bj x C x βx χ j x. Since q C log R n PR n and βx satisfy condition 2.4 and 2.5 which q instead of p. So, applying Proposition 2.2, emma 2.2, 2.7, 3.6 and 3.7, we obtain x γx Hβ f χ k q 2 Rn f j χ q Rn k + x γx βx n χ R n j j=k+ j=k+ j=k+ j=k+ j=k+ f j χ q Rn k 2 jn R n + x γx I β χ Bj f j q Rn χ B k Rn 2 jn χ Bj q 2 R n χ Bk R n f j q R n χ Bj q 2 R n 2 k jnδ2 f j q R n. q R n q R n

12 434 JIANGONG WU Because of 0 < p /p 2, therefore, applying 3.4, and combining 3.5 and 3.8, and using α > λ nδ 2, it follows that + x γx H β f p M p 2, Rn k0 sup 2 k0λp 2 kαp p 2 k jnδ2 f j q k R 0 Z n k= j=k+ f p M sup k0 2 k0λp 2 kαp 2 k jnδ2 2 jλ α p p,q Rn k 0 Z k= j=k+ f p M sup k0 2 k0λp 2 kλp 2 k jnδ2+α λ p p,q Rn k 0 Z k= j=k+ f p M sup k0 2 k0λp 2 kλp p,q Rn k 0 Z k= Consequently, the proof of Theorem 3.2 is completed. f p M. p,q Rn In particular, when γx = 0 and β is constant exponent, the main results above are proved by Zhang and Wu in [9]. Acknowledgments. The author cordially thank the referees for their valuable suggestions and useful comments which have lead to the improvement of this paper. This work was partially supported by the Fund No of Heilongjiang Provincial Education Department and the Project No.SY2033 of Mudanjiang Normal University. References [] A. Almeida and D. Drihem, Maximal, potential and singular type operators on Herz spaces with variable exponents, J. Math. Anal. Appl , no. 2, [2] M. Christ and. Grafakos, Best constants for two nonconvolution inequalities, Proc. Amer. Math. Soc , no. 6, [3] Z. Fu, Z. iu, S. u, and H. Wang, Characterization for commutators of n-dimensional fractional Hardy operators, Sci. China Ser. A , no. 0, [4] J.. Wu, Boundedness of multilinear commutators of fractional Hardy operators, Acta Math. Sci. Ser. A Chin. Ed. 3 20, no. 4, [5] J.. Wu and Q. G. iu, λ-central BMO estimates for higher order commutators of Hardy operators, Commun. Math. Res. In Prss. [6] J.. Wu and J. M. Wang, Boundedness of multilinear commutators of fractional Hardy operators, Appl. Math. J. Chinese Univ. Ser. A , no., 5 2. [7] J.. Wu and P. Zhang, Boundedness of commutators of the fractional Hardy operators on Herz-Morrey spaces with variable exponent, Adv. Math. China, In Press. [8], Boundedness of multilinear Hardy type operators on product of Herz-Morrey spaces with variable exponent, Appl. Math. J. Chinese Univ. Ser. A , no. 2, [9] P. Zhang and J.. Wu, Boundedness of fractional Hardy type operators on Herz-Morrey spaces with variable exponent, J. Math. Practice Theory , no. 7,

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