Jordan Journal of Mathematics and Statistics (JJMS 9(1, 2016, pp 17-30 BOUNDEDNESS OF COMMUTATORS ON HERZ-TYPE HARDY SPACES WITH VARIABLE EXPONENT WANG HONGBIN Abstract. In this paper, we obtain the boundedness of some commutators generated by the Calderón-Zygmund singular integral operator, the Littlewood-Paley operator and BMO functions on Herz-type Hardy spaces with variable exponent. 1. Introduction The theory of function spaces with variable exponent has extensively studied by researchers since the work of Kováčik and Rákosník [4] appeared in 1991. In [1] and [8], the authors proved the boundedness of some integral operators on variable L p spaces, respectively. In addition, the authors defined the Herz-type Hardy spaces with variable exponent and gave their atomic characterizations in [9]. Motivated by [1], [5] and [7], we will study the boundedness of some commutators generated by the Calderón-Zygmund singular integral operator, the Littlewood-Paley operator and BMO functions on the Herz-type Hardy space with variable exponent. Given an open set Ω R n, and a measurable function p( : Ω [1,, L p( (Ω denotes the set of measurable functions f on Ω such that for some λ > 0, ( p(x f(x dx <. Ω λ 2000 Mathematics Subject Classification. 42B20, 42B35. Key words and phrases. Herz-type Hardy space, variable exponent, commutator. Copyright c Deanship of Research and Graduate Studies, Yarmouk University, Irbid, Jordan. Received: June 14, 2015 Accepted: Oct. 26, 2015. 17
18 WANG HONGBIN This set becomes a Banach function space when equipped with the Luxemburg- Nakano norm f L p( (Ω = inf λ > 0 : Ω ( p(x f(x dx 1. λ These spaces are referred to as variable L p spaces, since they generalized the standard L p spaces: if p(x = p is constant, then L p( (Ω is isometrically isomorphic to L p (Ω. For all compact subsets E Ω, the space L p( loc (Ω is defined by Lp( loc (Ω := f : f L p( (E. Define P(Ω to be set of p( : Ω [1, such that p = ess infp(x : x Ω > 1, p + = ess supp(x : x Ω <. Denote p (x = p(x/(p(x 1. Let B(Ω be the set of p( P(Ω such that the Hardy-Littlewood maximal operator M is bounded on L p( (Ω. In variable L p spaces there are some important lemmas as follows. Lemma 1.1. ([4] Let p( P(Ω. If f L p( (Ω and g L p ( (Ω, then fg is integrable on Ω and Ω f(xg(x dx r p f L p( g L p (, where r p = 1 + 1/p 1/p +. This inequality is named the generalized Hölder inequality with respect to the variable L p spaces. Lemma 1.2. ([2] Let p( B(R n. Then there exists a positive constant C such that for all balls B in R n and all measurable subsets S B, χ B L p( (R n χ S L p( (R n C B S, χ S Lp( (Rn χ B L p( (R n C ( δ1 S, B
BOUNDEDNESS OF COMMUTATORS 19 and χ S L p ( (R n χ B L p ( (R n C ( δ2 S, B where δ 1, δ 2 are constants with 0 < δ 1, δ 2 < 1 and χ S, χ B are the characteristic functions of S, B respectively. Throughout this paper δ 2 is the same as in Lemma 1.2. Lemma 1.3. ([2] Suppose p( B(R n. Then there exists a constant C > 0 such that for all balls B in R n, that 1 B χ B L p( (R n χ B L p ( (R n C. Recall that the space BMO(R n consists of all locally integrable functions f such f = sup Q 1 f(x f Q dx <, Q Q where f Q = Q 1 Q f(ydy, the supremum is taken over all cubes Q Rn with sides parallel to the coordinate axes and Q denotes the Lebesgue measure of Q. Lemma 1.4. ([3] Let p( B(R n, k be a positive integer and B be a ball in R n. Then we have that for all b BMO(R n and all j, i Z with j > i, 1 C b k sup B:ball 1 (b b B k χ B χ B L p( (R n C b k, L p( (R n (b b Bi k χ Bj L p( (R n C(j i k b k χ Bj L p( (R n, where B i = x R n : x 2 i and B j = x R n : x 2 j. Next we recall the definition of the Herz-type spaces with variable exponent. Let B k = x R n : x 2 k and A k = B k \ B k 1 for k Z. Denote Z + and N as the sets of all positive and non-negative integers, χ k = χ Ak for k Z, χ k = χ k if k Z + and χ 0 = χ B0.
20 WANG HONGBIN Definition 1.1. ([2] Let α R, 0 < p and q( P(R n. The homogeneous Herz space with variable exponent where K q( (Rn is defined by K q( (Rn = f L q( loc (Rn \ 0 : f K q( (Rn <, f K q( (Rn = 2 kαp fχ k p L q( (R n 1/p. The non-homogeneous Herz space with variable exponent K q( (Rn is defined by K q( (Rn = f L q( loc (Rn : f K q( (Rn <, where f K q( (Rn = 1/p 2 kαp f χ k p L q( (R. n k=0 In [9], the authors gave the definition of Herz-type Hardy space with variable exponent and the atomic decomposition characterizations. S(R n denotes the space of Schwartz functions, and S (R n denotes the dual space of S(R n. Let G N (f(x be the grand maximal function of f(x defined by G N (f(x = sup φ A N φ (f(x, where A N = φ S(R n : sup x α D β φ(x 1 and N > n + 1, φ α, β N nontangential maximal operator defined by is the with φ t (x = t n φ(x/t. φ (f(x = sup φ t f(y y x <t Definition 1.2. ([9] Let α R, 0 < p <, q( P(R n and N > n + 1. (i The homogeneous Herz-type Hardy space with variable exponent H defined by H K q( (Rn = f S (R n : G N (f(x K q( (Rn K q( (Rn is
BOUNDEDNESS OF COMMUTATORS 21 and f H K q( (Rn = G N(f K q( (Rn. (ii The non-homogeneous Herz-type Hardy space with variable exponent HK q( (Rn is defined by HK q( (Rn = f S (R n : G N (f(x K q( (Rn and f HK q( (Rn = G N (f K q( (Rn. For x R we denote by [x] the largest integer less than or equal to x. Similar to [9], we have Definition 1.3. Let nδ 2 α <, q( P(R n, s N and b L q ( loc (Rn. (i A function a(x on R n is said to be a central (α, q(, s; b-atom, if it satisfies (1 supp a B(0, r = x R n : x < r. (2 a L q( (R n B(0, r α/n. (3 R n a(xx β dx = R n a(xb(xx β dx = 0, β s. (ii A function a(x on R n is said to be a central (α, q(, s; b-atom of restricted type, if it satisfies the conditions (2, (3 above and (1 supp a B(0, r, r 1. If r = 2 k for some k Z in Definition 1.3, then the corresponding central (α, q(, s; b-atom is called a dyadic central (α, q(, s; b-atom. Lemma 1.5. Let nδ 2 α <, 0 < p < and q( B(R n. Then f H K q( (Rn (or HK q( (Rn if and only if f = λ k a k (or λ k a k, in the sense of S (R n, k=0
22 WANG HONGBIN where each a k is a central (α, q(, s; b-atom(or central (α, q(, s; b-atom of restricted type with support contained in B k and λ k p < (or λ k p <. Moreover, f H K q( (Rn inf ( 1/p λ k p or k=0 f HK q( (Rn inf where the infimum is taken over all above decompositions of f. ( k=0 1/p λ k p, 2. Boundedness of the commutator of the Calderón-Zygmund singular integral operator Let b BMO(R n and let T be the Calderón-Zygmund singular integral operator of Coifman and Meyer, that is, T f(x = K(x, yf(ydy, R n x supp f, with the kernel K(x, y : R n R n \ x = y satisfying K(x, y C x y n, y z ε K(x, y K(x, z C x y n+ε, 2 y z < x y, x w ε K(x, y K(w, y C x y n+ε, 2 x w < x y, for some ε (0, 1]. The commutator [b, T ] generated by b and T is defined by [b, T ]f(x = b(xt f(x T (bf(x. In [1], the authors proved that the commutator [b, T ] is bounded on L p( (R n for p( B(R n. In this section we will generalize the result to the case of Herz-type Hardy spaces with variable exponent. Theorem 2.1. Suppose q( B(R n, 0 < p <, nδ 2 BMO(R n. Then the commutator [b, T ] is bounded from H to K q( (Rn (or K q( (Rn. α < nδ 2 + ε and b K q( (Rn (or HK q( (Rn
BOUNDEDNESS OF COMMUTATORS 23 Proof. We only prove homogeneous case. The non-homogeneous case can be proved in the same way. Let f H K q( (Rn. By Lemma 1.5, we get f = λ j a j, where f H K q( (Rn inf( λ j p 1/p (the infimum is taken over above decompositions of f, and a j is a dyadic central (α, q(, 0; b-atom with the support B j. We have (2.1 [b, T ]f p K q( (Rn = C +C 2 kαp χ k [b, T ]f p L q( (R n =: E 1 + E 2. ( k 2 2 kαp 2 kαp ( j=k 1 λ j χ k [b, T ]a j L q( (R n λ j χ k [b, T ]a j L q( (R n p p have Now we estimate E 2, by the L q( (R n -boundedness of the commutator [b, T ] we (2.2 E 2 ( 2 kαp λ j a j L q( (R n ( j=k 1 p λ j 2 (k jα j=k 1 λ j p. p Let us now estimate E 1. For each j k 2, we first compute χ k [b, T ]a j L q( (R. n Let b Bj be the mean value of b on the ball B j. Note that if x A k, y B j and j k 2, then 2 y < x. Using the vanishing moments of a j and the generalized
24 WANG HONGBIN Hölder inequality we have [b, T ]a j (x b(x b Bj k(x, y k(x, 0 a j (y dy + k(x, y k(x, 0 b Bj b(y a j (y dy B j B j C2 b(x jε k(n+ε b Bj a j (y dy + b Bj b(y a j (y dy B j B j C2 jε k(n+ε a j L q( (R n b(x b Bj χ Bj L q ( (R n + (b B j bχ Bj L q ( (R n. So by Lemma 1.1-1.4 we have χ k [b, T ]a j L q( (R n C2 jε k(n+ε a j L q( (R n (b b Bj χ k L q( (R χ n Bj L q ( (R n + (b B j bχ Bj L q ( (R n χ k L q( (R n C2 jε k(n+ε a j L q( (R n (k j b χ Bk L q( (R χ n Bj L q ( (R n + b χ Bj L q ( (R n χ B k L q( (R n C2 jε k(n+ε a j L q( (R (k j b n χ Bk L q( (R χ n Bj L q ( (R n χ C(k j2 (j kε Bj L q b a j ( (R n L q( (R n χ Bk L q ( (R n C(k j2 jα+(j k(ε+nδ 2 b. Thus we obtain E 1 = C b p ( k 2 2 kαp ( k 2 λ j (k j2 jα+(j k(ε+nδ 2 p λ j (k j2 2 α (j k(ε+nδ. p
BOUNDEDNESS OF COMMUTATORS 25 When 1 < p <, take 1/p+1/p = 1. Since ε+nδ 2 α > 0, by the Hölder inequality we have (2.3 E 1 = C b p ( k 2 ( k 2 λ j p 2 (j k(ε+nδ 2 αp/2 (k j p 2 (j k(ε+nδ 2 αp /2 ( k 2 λ j p ( λ j p. λ j p 2 (j k(ε+nδ 2 αp/2 k=j+2 2 (j k(ε+nδ 2 αp/2 p/p When 0 < p 1, we have (2.4 E 1 = C b p ( k 2 λ j p (k j p 2 (j k(ε+nδ 2 αp ( λ j p (k j p 2 (j k(ε+nδ 2 αp λ j p. k=j+2 Therefore, by (2.1-(2.4 we complete the proof of Theorem 2.1. 3. Boundedness of the commutator of the Littlewood-Paley operator Let ε > 0 and fix a function ψ satisfying the following properties: (1 ψ(xdx = 0, R n (2 ψ(x C(1 + x (n+ε, (3 ψ(x + y ψ(x C y ε (1 + x (n+1+ε when 2 y < x.
26 WANG HONGBIN The Littlewood-Paley operator is defined by ( S ψ f(x = Γ(x f ψ t (y 2 dydt 1/2, t n+1 where Γ(x = (y, t R n+1 + : x y < t and ψ t (x = t n ψ(x/t for t > 0. Let b be a locally integrable function. The commutator of the Littlewood-Paley operator is defined by ( [b, S ψ ]f(x = Γ(x F b,t (f(x, y 2dydt 1/2, t n+1 where F b,t (f(x, y = ψ t (y zf(z(b(x b(zdz. R n In [1], the authors proved that the operator S ψ is bounded on L p( (R n for p( B(R n. Similar to the method of [1], we can easily obtain that the commutator [b, S ψ ] is also bounded on L p( (R n for p( B(R n and b BMO(R n. In this section we will generalize the result to the case of Herz-type Hardy spaces with variable exponent. Theorem 3.1. Suppose q( B(R n, 0 < p <, nδ 2 BMO(R n. Then the commutator [b, S ψ ] is bounded from H to K q( (Rn (or K q( (Rn. α < nδ 2 + ε and b K q( (Rn (or HK q( (Rn Proof. We only prove homogeneous case. The non-homogeneous case can be proved in the same way. Let f H K q( (Rn. By Lemma 1.5, we get f = λ j a j, where f H K q( (Rn inf( λ j p 1/p (the infimum is taken over above decompositions of f, and a j is a dyadic central (α, q(, 0; b-atom with the support B j. We have
BOUNDEDNESS OF COMMUTATORS 27 (3.1 [b, S ψ ]f p K q( (Rn = C +C 2 kαp χ k [b, S ψ ]f p L q( (R n =: F 1 + F 2. ( k 2 2 kαp 2 kαp ( j=k 1 λ j χ k [b, S ψ ]a j L q( (R n λ j χ k [b, S ψ ]a j L q( (R n p p have Now we estimate F 2, by the L q( (R n -boundedness of the commutator [b, S ψ ] we (3.2 F 2 ( 2 kαp λ j a j L q( (R n ( j=k 1 p λ j 2 (k jα j=k 1 λ j p. p Let us now estimate F 1. Similar to the proof of Theorem 2.1, for each j k 2, we first compute χ k [b, S ψ ]a j L q( (R n. Let b Bj = B j 1 B j b(xdx. Note that if x A k, y B j and j k 2, then 2 y < x. Using the vanishing moments of a j
28 WANG HONGBIN and the generalized Hölder inequality we have [b, S ψ ]a j (x ( 2 1/2 dydt ψ t (y z ψ t (y 0 a j (z b(x b(z dz Γ(x B j t n+1 ( 2 1/2 C t n ( z /t ε dydt Γ(x B j (1 + y /t n+1+ε a j(z b(x b(z dz t n+1 ( 2 1/2 C2 jε t dydt Γ(x B j (t + y n+1+ε a j(z b(x b(z dz t n+1 [ ] C2 jε t 1 n 1/2 dydt a Γ(x (t + y 2(n+1+ε j (z b(x b(z dz B [ j ] 1/2 C2 jε t dt a 0 (t + x 2(n+1+ε j (z b(x b(z dz B j C2 b(x jε k(n+ε b Bj a j (z dz + b Bj b(z a j (z dz B j B j C2 jε k(n+ε a j L q( (R n b(x b Bj χ Bj L q ( (R n + (b B j bχ Bj L q ( (R n. So by Lemma 1.1-1.4 we have χ k [b, S ψ ]a j L q( (R n C2 jε k(n+ε a j L q( (R n (b b Bj χ k L q( (R χ n Bj L q ( (R n + (b B j bχ Bj L q ( (R n χ k L q( (R n C2 jε k(n+ε a j L q( (R n (k j b χ Bk L q( (R χ n Bj L q ( (R n + b χ Bj L q ( (R n χ B k L q( (R n C2 jε k(n+ε a j L q( (R (k j b n χ Bk L q( (R χ n Bj L q ( (R n χ C(k j2 (j kε Bj L q b a j ( (R n L q( (R n χ Bk L q ( (R n C(k j2 jα+(j k(ε+nδ 2 b.
BOUNDEDNESS OF COMMUTATORS 29 Thus we obtain F 1 = C b p ( k 2 2 kαp ( k 2 λ j (k j2 jα+(j k(ε+nδ 2 λ j (k j2 (j k(ε+nδ 2 α p. p When 1 < p <, take 1/p+1/p = 1. Since ε+nδ 2 α > 0, by the Hölder inequality we have (3.3 F 1 = C b p ( k 2 ( k 2 λ j p 2 (j k(ε+nδ 2 αp/2 (k j p 2 (j k(ε+nδ 2 αp /2 ( k 2 λ j p ( λ j p. λ j p 2 (j k(ε+nδ 2 αp/2 k=j+2 2 (j k(ε+nδ 2 αp/2 p/p When 0 < p 1, we have (3.4 F 1 = C b p ( k 2 λ j p (k j p 2 (j k(ε+nδ 2 αp ( λ j p (k j p 2 (j k(ε+nδ 2 αp λ j p. k=j+2 Therefore, by (3.1-(3.4 we complete the proof of Theorem 3.1.
30 WANG HONGBIN Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 11171345. The author is very grateful to the referees for their valuable comments. References [1] D. Cruz-Uribe, SFO, A. Fiorenza, J. M. Martell and C. Pérez, The boundedness of classical operators on variable L p spaces, Ann. Acad. Sci. Fen. Math. 31(2006, 239 264 [2] M. Izuki, Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization, Anal. Math. 36(2010, 33 50 [3] M. Izuki, Boundedness of commutators on Herz spaces with variable exponent, Rend. del Circolo Mate. di Palermo 59(2010, 199 213 [4] O. Kováčik and J. Rákosník, On spaces L p(x and W k,p(x, Czechoslovak Math. J. 41(1991, 592 618 [5] L. Z. Liu, Continuity for commutators of Littlewood-Paly operators on Certain Hardy spaces, J. Korean Math. Soc. 40(2003, 41 60 [6] Z. G. Liu, H. B. Wang, Boundedness of Marcinkiewicz integrals on Herz spaces with variable exponent, Jordan J. Math. Stat. 4(2012, 223 239 [7] S. Z. Lu, D. C. Yang, The continuity of commutators on Herz-type spaces, Michigan Math. J. 44(1997, 255 281 [8] H. B. Wang, Z. W. Fu, Z. G. Liu, Higher-order commutators of Marcinkiewicz integrals on variable Lebesgue spaces, Acta Math. Sci. Ser. A Chin. Ed. 32(2012, 1092 1101 [9] H. B. Wang, Z. G. Liu, The Herz-type Hardy spaces with variable exponent and its applications, Taiwanese J. Math. 16(2012, 1363 1389 [10] H. B. Wang, Z. G. Liu, Some characterizations of Herz-type Hardy spaces with variable exponent, Ann. Funct. Anal. 6(2015, 224 233 School of Science, Shandong University of Technology, Zibo 255000, Shandong, China E-mail address: hbwang 2006@163.com