Jordan Journal of Mathematics and Statistics (JJMS) 5(4), 2012, pp

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1 Jordan Journal of Mathematics and Statistics (JJMS) 5(4), 2012, pp BOUNDEDNESS OF MARCINKIEWICZ INTEGRALS ON HERZ SPACES WITH VARIABLE EXPONENT ZONGGUANG LIU (1) AND HONGBIN WANG (2) Abstract In this paper, the authors obtain some boundedness for Marcinkiewicz integrals and their commutators on Herz spaces with variable exponent 1 Introduction 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 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) The L p spaces with variable exponent are a special case of Musielak-Orlicz spaces 2000 Mathematics Subject Classification 42B20, 42B35 Key words and phrases Herz spaces, variable exponent, Marcinkiewicz integral, commutator Copyright c Deanship of Research and Graduate Studies, Yarmouk University, Irbid, Jordan Received: July 17, 2011 Accepted : Nov28,

2 224 ZONGGUANG LIU AND HONGBIN WANG For all compact subsets E Ω, the space L p( ) loc (Ω) is defined by L p( ) loc (Ω) := f : f Lp( ) (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 addition, we denote the Lebesgue measure and characteristic function of measurable set A R n by A and χ A respectively In variable L p spaces there are some important lemmas as follows Lemma 11 ([3]) Let p( ) P(Ω) If f L p( ) (Ω) and g L p ( ) (Ω), then fg is integrable on Ω and where Ω f(x)g(x) dx r p f L p( )(Ω) g L p ( )(Ω), r p = 1 + 1/p 1/p + This inequality is named the generalized Hölder inequality with respect to the variable L p spaces Lemma 12 ([1]) 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 ) B S, χ S L p( ) (R n ) χ B L p( ) (R n ) ( ) δ1 S, χ S L p ( ) (R n ) B χ B L p ( ) (R n ) where δ 1, δ 2 are constants with 0 < δ 1, δ 2 < 1 ( ) δ2 S, B Throughout this paper δ 1 and δ 2 are the same as in Lemma 12

3 BOUNDEDNESS OF MARCINKIEWICZ INTEGRALS 225 Lemma 13 ([1]) Suppose p( ) B(R n ) Then there exists a constant C > 0 such that for all balls B in R n, 1 B χ B L p( ) (R n ) χ B L p ( ) (R n ) Firstly we recall the definition of the Herz 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 Definition 11 ([1]) Let α R, 0 < p and q( ) P(R n ) The homogeneous Herz space K α,p q( ) (Rn ) is defined by where K α,p q( ) (Rn ) = f L q( ) loc (Rn \ 0) : f K α,p q( ) (Rn ) <, f K α,p q( ) (Rn ) = 2 kαp fχ k p L q( ) (R n ) The non-homogeneous Herz space K α,p q( ) (Rn ) is defined by K α,p q( ) (Rn ) = f L q( ) loc (Rn ) : f K α,p q( ) (Rn ) <, where f K α,p q( ) (Rn ) = 1/p 2 kαp f χ k p L q( ) (R ) n k=0 Suppose that S n 1 denote the unit sphere in R n (n 2) equipped with normalized Lebesgue measure Let Ω Lip β (R n ) for 0 < β 1 be homogeneous function of degree zero and S n 1 Ω(x )dσ(x ) = 0,

4 226 ZONGGUANG LIU AND HONGBIN WANG where x = x/ x for any x 0 In 1958, Stein [5] introduced the Marcinkiewicz integral related to the Littlewood-Paley g function on R n as follows ( µ(f)(x) = 0 F t (f)(x) 2 dt, t 3 where F t (f)(x) = f(y)dy x y n 1 It was shown that µ is of type (p, p) for 1 < p 2 and of weak type (1, 1) Let b be a locally integrable function on R n, the commutator generated by the Marcinkiewicz integral µ and b is defined by ( [b, µ](f)(x) = 0 ) [b(x) b(y)]f(y)dy 2 1/2 dt x y n 1 t 3 Recall that the space BMO(R n ) consists of all locally integrable functions f such that f = sup Q 1 f(x) f Q dx <, Q Q where f Q = Q 1 Q f(y)dy, the supremum is taken over all cubes Q Rn with sides parallel to the coordinate axes Lemma 14 ([2]) Let 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 1 (b b B ) k χ B χ B L p( ) (R n ) b k, L p( ) (R n ) (b b Bi ) k χ Bj L p( ) (R n ) (j i) k b k χ Bj L p( ) (R n ) There are two lemmas for µ and [b, µ] respectively

5 BOUNDEDNESS OF MARCINKIEWICZ INTEGRALS 227 Lemma 15 ([6]) If p( ) B(R n ), then there exists a constant C such that for any f L p( ) (R n ), µ(f) L p( ) (R n ) f L p( ) (R n ) Lemma 16 ([6]) If p( ) B(R n ) and b BMO(R n ), then [b, µ](f) L p( ) (R n ) b f L p( ) (R n ) Inspired by [1-6], we obtain some boundedness for Marcinkiewicz integrals and their commutators on Herz spaces with variable exponent 2 Boundedness of Marcinkiewicz integral operators In this section we will prove the boundedness on Herz spaces with variable exponent for Marcinkiewicz integral operators µ Theorem 21 Suppose q( ) B(R n ), 0 < p and nδ 1 < α < nδ 2 Then µ is bounded on K α,p q( ) (Rn ) and K α,p q( ) (Rn ) Proof It suffices to prove the homogeneous case The non-homogeneous case can be proved in the same way We suppose 0 < p <, since the proof of the case p = α,p is easier Let f K q( ) (Rn ), and we write f(x) = fχ j (x) = f j (x) Then we have

6 228 ZONGGUANG LIU AND HONGBIN WANG (21) µ(f) K α,p q( ) (Rn ) = +C +C 2 kαp µ(f)χ k p L q( ) (R n ) ( k 2 2 kαp ( k+1 2 kαp j=k 1 2 kαp ( =: CE 1 + CE 2 + CE 3 µ(f j )χ k L q( ) (R n ) µ(f j )χ k L q( ) (R n ) µ(f j )χ k L q( ) (R n ) ) p ) p ) p By Lemma 15, we know that µ is bounded on L q( ) (R n ) So we have E 2 2 kαp f k p L q( ) (R n ) = C f K α,p q( ) (Rn ) Now we estimate E 1 We consider µ(f j )(x) ( x 0 ( + x =: F 1 + F 2 x y f j(y)dy 2 dt n 1 t 3 x y f j(y)dy 2 dt n 1 t 3 Note that x A k, y A j and j k 2 So we know that x y x, and by mean value theorem we have (22) 1 x y 1 2 x 2 y x y 3

7 BOUNDEDNESS OF MARCINKIEWICZ INTEGRALS 229 Since Ω is bounded, by (22), the Minkowski inequality and the generalized Hölder inequality we have F 1 R n R n R n 2j/2 x n+1/2 ( f j (y) x x y n 1 x y f j (y) 1 x y n 1 f j (y) y 1/2 x y n 1 x y A j f(y) dy dt dy t 3 x y 1 2 x 2 3/2 dy 1/2 2 (j k)/2 2 kn f j L q( ) (R n ) χ j L q ( ) (R n ) dy Similarly, we consider F 2 Noting that x y x, by the Minkowski inequality and the generalized Hölder inequality we have F 2 R n R n ( f j (y) x y n 1 x f j (y) x y dy n dt dy t 3 2 kn f j L q( ) (R n ) χ j L q ( ) (R n ) So we have µ(f j )(x) 2 kn f j L q( ) (R n ) χ j L q ( ) (R n ) By Lemma 12 and Lemma 13 we have µ(f j )χ k L q( ) (R n ) 2 kn f j L q( ) (R n ) χ j L q ( ) (R n ) χ k L q( ) (R n ) 2 kn f j L q( ) (R ) χ n Bj L q ( ) (R n ) χ B k L q( ) (R n ) ( ) 2 kn f j L q( ) (R ) χ n Bj L q ( ) (R n ) B k χ Bk 1 L q ( ) (R n ) χ Bj L q f j ( ) (R n ) L q( ) (R n ) χ Bk L q ( ) (R n ) 2 (j k)nδ 2 f j L q( ) (R n )

8 230 ZONGGUANG LIU AND HONGBIN WANG Thus we obtain E 1 = C ( k 2 2 kαp ( k 2 2 (j k)nδ 2 f j L q( ) (R n ) ) p 2 jα 2 (j k)(nδ 2 α) f j L q( ) (R n )) p If 1 < p <, take 1/p + 1/p = 1 Since nδ 2 α > 0, by the Hölder inequality we have (23) E 1 = C ( k 2 ( k 2 2 (j k)(nδ 2 α)p /2 ( k 2 = C f K α,p q( ) (Rn ) ) 2(j k)(nδ 2 α)p/2 ) p/p 2(j k)(nδ 2 α)p/2 ( k=j+2 2 (j k)(nδ 2 α)p/2 ) ) If 0 < p 1, then we have (24) E 1 = C k 2 f K α,p q( ) (Rn ) 2 jαp 2 (j k)(nδ 2 α)p f j p L q( ) (R n ) ( k=j+2 2 (j k)(nδ 2 α)p )

9 BOUNDEDNESS OF MARCINKIEWICZ INTEGRALS 231 Let us now estimate E 3 Note that x A k, y A j and j k + 2, so we have x y y We consider ( y ) µ(f j )(x) 0 x y f j(y)dy 2 1/2 dt n 1 t 3 ( ) + x y f j(y)dy 2 1/2 dt n 1 t 3 =: G 1 + G 2 Similar to the estimate for F 1, we get y G 1 2 (k j)/2 2 jn f j L q( ) (R n ) χ j L q ( ) (R n ) Similar to the estimate for F 2, we get G 2 2 jn f j L q( ) (R n ) χ j L q ( ) (R n ) So we have By Lemma 12 and Lemma 13 we have µ(f j )(x) 2 jn f j L q( ) (R n ) χ j L q ( ) (R n ) µ(f j )χ k L q( ) (R n ) 2 jn f j L q( ) (R n ) χ j L q ( ) (R n ) χ k L q( ) (R n ) 2 jn f j L q( ) (R ) χ n Bj L q ( ) (R n ) χ B k L q( ) (R n ) ( ) 2 jn f j L q( ) (R n ) B j χ Bj 1 χ L q( ) (R n ) Bk L q( ) (R n ) χ Bk L f j q( ) (R n ) L q( ) (R n ) χ Bj L q( ) (R n ) 2 (k j)nδ 1 f j L q( ) (R n ) Thus we obtain E 3 ( ) p 2 kαp 2 (k j)nδ 1 f j L q( ) (R n ) = C ( p 2 jα 2 (k j)(nδ1+α) f j L q( ) (R )) n

10 232 ZONGGUANG LIU AND HONGBIN WANG If 1 < p <, take 1/p + 1/p = 1 Since nδ 1 + α > 0, by the Hölder inequality we have (25) E 3 ( = C ( 2 (k j)(nδ 1+α)p /2 ( = C f K α,p q( ) (Rn ) If 0 < p 1, then we have (26) E 3 = C ) 2(k j)(nδ 1+α)p/2 ) p/p 2(k j)(nδ 1+α)p/2 f K α,p q( ) (Rn ) ( j 2 2 (k j)(nδ 1+α)p/2 2 jαp 2 (k j)(nδ 1+α)p f j p L q( ) (R n ) ( j 2 2 (k j)(nδ 1+α)p ) ) ) Therefore, by (21), (23)-(26) we complete the proof of Theorem 21 3 Boundedness of the commutators of Marcinkiewicz integral operators In this section we will prove the boundedness on Herz spaces with variable exponent for the commutators of Marcinkiewicz integral operators [b, µ] Theorem 31 Suppose b BMO(R n ), q( ) B(R n ), 0 < p and nδ 1 < α < nδ 2 Then [b, µ] is bounded on K α,p q( ) (Rn ) and K α,p q( ) (Rn )

11 BOUNDEDNESS OF MARCINKIEWICZ INTEGRALS 233 Proof Similar to Theorem 21, we only prove homogeneous case and still suppose α,p 0 < p < Let f K q( ) (Rn ), and we write f(x) = fχ j (x) = f j (x) Then we have (31) [b, µ](f) K α,p q( ) (Rn ) = +C +C 2 kαp [b, µ](f)χ k p L q( ) (R n ) ( k 2 2 kαp ( k+1 2 kαp j=k 1 2 kαp ( =: CU 1 + CU 2 + CU 3 [b, µ](f j )χ k L q( ) (R n ) [b, µ](f j )χ k L q( ) (R n ) [b, µ](f j )χ k L q( ) (R n ) ) p ) p ) p By Lemma 16, we know that [b, µ] is bounded on L q( ) (R n ) So we have U 2 Now we estimate U 1 We consider 2 kαp f k p L q( ) (R n ) = C f K α,p q( ) (Rn ) [b, µ](f j )(x) ( x 0 ( + x =: V 1 + V 2 x y [b(x) b(y)]f j(y)dy 2 dt n 1 t 3 x y [b(x) b(y)]f j(y)dy 2 dt n 1 t 3 Note that x A k, y A j and j k 2, and we know that x y x Since Ω is bounded, by (22), the Minkowski inequality and the generalized Hölder inequality

12 234 ZONGGUANG LIU AND HONGBIN WANG we have V 1 ( b(x) b(y) f j (y) x dt dy R x y n n 1 x y t 3 b(x) b(y) f j (y) 1 R x y n n 1 x y 1 1/2 2 x 2 dy b(x) b(y) f j (y) y 1/2 dy R x y n n 1 x y 3/2 2j/2 b(x) b(y) f(y) dy x n+1/2 A j 2 (j k)/2 2 b(x) kn b Bj f j (y) dy + b Bj b(y) f j (y) dy A j A j 2 (j k)/2 2 kn f j L q( ) (R n ) b(x) b Bj χ Bj L q ( ) (R n ) + (b B j b)χ j L q ( ) (R n ) Similarly, we consider V 2 Noting that x y x, by the Minkowski inequality and the generalized Hölder inequality we have V 2 ( b(x) b(y) f j (y) dt dy R x y n n 1 x t 3 b(x) b(y) f j (y) dy R x y n n 2 kn f j L q( ) (R n ) b(x) b Bj χ Bj L q ( ) (R n ) + (b B j b)χ j L q ( ) (R n ) So we have [b, µ](f j )(x) 2 kn f j L q( ) (R n ) b(x) b Bj χ Bj L q ( ) (R n ) + (b B j b)χ j L q ( ) (R n )

13 BOUNDEDNESS OF MARCINKIEWICZ INTEGRALS 235 By Lemma 12, Lemma 13 and Lemma 14 we have [b, µ](f j )χ k L q( ) (R n ) 2 kn f j L q( ) (R n ) (b b Bj )χ k L q( ) (R ) χ n Bj L q ( ) (R n ) + (b B j b)χ j L q ( ) (R n ) χ k L q( ) (R n ) 2 kn f 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 ) 2 kn f j L q( ) (R )(k j) b n χ Bk L q( ) (R ) χ n Bj L q ( ) (R n ) χ Bj L q (k j) b f j ( ) (R n ) L q( ) (R n ) χ Bk L q ( ) (R n ) 2 (j k)nδ 2 (k j) b f j L q( ) (R n ) Thus we obtain U 1 = C ( k 2 2 kαp ( k 2 2 (j k)nδ 2 (k j) b f j L q( ) (R n ) ) p 2 jα 2 (j k)(nδ 2 α) (k j) b f j L q( ) (R n )) p

14 236 ZONGGUANG LIU AND HONGBIN WANG If 1 < p <, take 1/p + 1/p = 1 Since nδ 2 α > 0, by the Hölder inequality we have (32) U 1 b ( k 2 b ( k 2 ) 2(j k)(nδ 2 α)p/2 2 (j k)(nδ 2 α)p /2 (k j) p ) p/p = C b b ( k 2 = C b f K α,p q( ) (Rn ) 2(j k)(nδ 2 α)p/2 ( k=j+2 2 (j k)(nδ 2 α)p/2 ) ) If 0 < p 1, then we have (33) U 1 b = C b k 2 b f K α,p q( ) (Rn ) 2 jαp 2 (j k)(nδ 2 α)p (k j) p f j p L q( ) (R n ) ( k=j+2 2 (j k)(nδ 2 α)p (k j) p ) Let us now estimate U 3 Note that x A k, y A j and j k + 2, so we have x y y We consider [b, µ](f j )(x) ( y 0 ( + y =: W 1 + W 2 x y [b(x) b(y)]f j(y)dy 2 dt n 1 t 3 x y [b(x) b(y)]f j(y)dy 2 dt n 1 t 3

15 BOUNDEDNESS OF MARCINKIEWICZ INTEGRALS 237 Similar to the estimate for V 1, we get W 1 2 (k j)/2 2 jn f j L q( ) (R n ) b(x) b Bj χ Bj L q ( ) (R n ) + (b B j b)χ j L q ( ) (R n ) Similar to the estimate for V 2, we get W 2 2 jn f j L q( ) (R n ) b(x) b Bj χ Bj L q ( ) (R n ) + (b B j b)χ j L q ( ) (R n ) So we have [b, µ](f j )(x) 2 jn f j L q( ) (R n ) b(x) b Bj χ Bj L q ( ) (R n ) + (b B j b)χ j L q ( ) (R n ) By Lemma 12, Lemma 13 and Lemma 14 we have [b, µ](f j )χ k L q( ) (R n ) 2 jn f j L q( ) (R n ) (b b Bj )χ k L q( ) (R ) χ n Bj L q ( ) (R n ) + (b B j b)χ j L q ( ) (R n ) χ k L q( ) (R n ) 2 jn f j L q( ) (R n ) b χ Bk L q( ) (R ) χ n Bj L q ( ) (R n ) + (j k) b χ Bj L q ( ) (R n ) χ B k L q( ) (R n ) 2 jn f j L q( ) (R )(j k) b n χ Bk L q( ) (R ) χ n Bj L q ( ) (R n ) χ Bk L (j k) b f j q( ) (R n ) L q( ) (R n ) χ Bj L q( ) (R n ) 2 (k j)nδ 1 (j k) b f j L q( ) (R n ) Thus we obtain U 3 ( ) p b 2 kαp 2 (k j)nδ 1 (j k) f j L q( ) (R n ) = C b ( p 2 jα 2 (k j)(nδ1+α) (j k) f j L q( ) (R )) n

16 238 ZONGGUANG LIU AND HONGBIN WANG If 1 < p <, take 1/p + 1/p = 1 Since nδ 1 + α > 0, by the Hölder inequality we have (34) U 3 b ( b ( ) 2(k j)(nδ 1+α)p/2 2 (k j)(nδ 1+α)p /2 (j k) p ) p/p = C b b ( = C b f K α,p q( ) (Rn ) 2(k j)(nδ 1+α)p/2 ( j 2 2 (k j)(nδ 1+α)p/2 ) ) If 0 < p 1, then we have (35) U 3 b = C b b f K α,p q( ) (Rn ) 2 jαp 2 (k j)(nδ 1+α)p (j k) p f j p L q( ) (R n ) ( j 2 2 (k j)(nδ 1+α)p (j k) p ) Therefore, by (31)-(35) we complete the proof of Theorem 31 Acknowledgement This work was supported by the National Natural Science Foundation of China (Grant Nos and ) and the Fundamental Research Funds for the Central Universities (Grant No 2010YS01) The authors are very grateful to the referees for their valuable comments

17 BOUNDEDNESS OF MARCINKIEWICZ INTEGRALS 239 References [1] M Izuki, Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization, Anal Math 36(2010), [2] M Izuki, Boundedness of commutators on Herz spaces with variable exponent, Rend del Circolo Mate di Palermo 59(2010), [3] O Kováčik and J Rákosník, On spaces L p(x) and W k,p(x), Czechoslovak Math J41(1991), [4] S Z Lu, D C Yang, The continuity of commutators on Herz-type spaces, Michigan Math J44(1997), [5] E M Stein, On the function of Littlewood-Paley, Lusin and Marcinkiewicz, Trans Amer Math Soc88(1958), [6] H B Wang, Z W Fu, Z G Liu, Higher-order commutators of Marcinkiewicz integrals on variable Lebesgue spaces, to appear (1) Department of Mathematics, China University of Mining and Technology(Beijing), Beijing , PR China address: liuzg@cumtbeducn (2) Department of Mathematical and Physical Science, Zibo Normall College, Zibo , Shandong, PRChina ; Department of Mathematics, China University of Mining and Technology(Beijing), Beijing , P R China address: hbwang 2006@163com (Corresponding author)