On the commutator of the Marcinkiewicz integral

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1 J. Math. Ana. A ) On the commutator of the Marcinkiewicz integra Guoen Hu a and Dunyan Yan b, a Deartment of Aied Mathematics, University of Information Engineering, P.O. Box , Zhengzhou 45000, Peoe s Reubic of China b Institute of Mathematics, Academy of Mathematics and Systems Sciences, Chinese Academy of Science, Beijing 00080, Peoe s Reubic of China Received 9 November 00 Submitted by R.H. Torres Abstract L R n ) boundedness is considered for the commutator of higher-dimensiona Marcinkiewicz integra. Some conditions imying the L R n ) and the L R n ) boundedness for the commutator of the Marcinkiewicz integra are obtained. 003 Pubished by Esevier Inc. Keywords: Marcinkiewicz integra; Commutator; Fourier transform estimate; Littewood Paey theory; BMOR n ). Introduction We wi work on R n, n. Let Ω be homogeneous of degree zero, integrabe on the unit shere S n and have mean vaue zero, i.e., S n Ωx)dx = 0. Define the Marcinkiewicz integra oerator µ Ω by µ Ω f )x) = 0 x y t ) Ωx y) fy)dy / x y n t 3. ) The research was suorted by the NSF of Henan Province and rofessor Xu Yuesheng s research grant in the rogram of One hundred Distinguished Young Scientists of the Chinese Academy of Sciences. * Corresonding author. E-mai addresses: huguoen@eyou.com G. Hu), ydunyan@amss.ac.cn D. Yan) X/03/$ see front matter 003 Pubished by Esevier Inc. doi:0.06/s00-47x0)

2 35 G. Hu, D. Yan / J. Math. Ana. A ) As we-known, this oerator was introduced by Stein [5], in order to generaize the onedimensiona Marcinkiewicz integra to higher-dimensiona case. Stein [5] showed that if Ω Li α S n ) for some 0 <α, then µ Ω is a bounded oerator on L R n ) for <, and a bounded maing from L R n ) to weak L R n ). Using the onedimensiona resut and Riesz transforms simiary as in the case of singuar integras see []) and interoation, Wash [8] roved that for each fixed <<, Ω Log L) /r og og L) /r ) S n ) is a sufficient condition such that µ Ω is bounded on L R n ),wherer = min{, } and = / ). Hu [3] showed that if Ω L q S n ) for some q>, then µ Ω is bounded on L R n,wx)dx) rovided that >q and w A /q or <<q and w / ) A /q,wherea denotes the weight function cass of Muckenhout see [6, Chater V] for the definition and roerties of A ). The urose of this aer is to estabish the L R n ) boundedness for the commutator of the oerator µ Ω.Forb BMOR n ) and ositive integer k, definef t;b,k by F t;b,k f )x) = x y t ) k Ωx y) bx) by) fy)dy. x y n The kth order commutator of the oerator µ Ω is defined by µ Ω;b,k f )x) = Ft;b,k f )x) ) / t 3. ) 0 This oerator was considered first by Torchinsky and Wang [7]. They showed that if Ω Li α S n ),thenµ Ω;b, is bounded on L R n ) for a <<. In this aer, we wi give some size condition on Ω imying the L R n ) boundedness of µ Ω;b,k for fixed <<. Our main resuts can be stated as foows. Theorem. Let Ω be homogeneous of degree zero and have mean vaue zero on the unit shere, k be a ositive integer. If Ω Log L) k+/ S n ), that is, Ωθ) og k+/ + ) Ωθ) dθ <. 3) S n Then for b BMOR n ), the commutator µ Ω;b,k defined by ) is bounded on L R n ) with bound C b k BMOR n ).Furthermore,ifΩ Log L)k+β S n ) for some / <β<, then the oerator µ Ω;b,k is bounded on L R n ) with bound C b k BMOR n ) rovided that /β < < / β). Remark. It seems that the method used in [8] does not ay to the commutator µ Ω;b,k.In this aer we wi use the technique invoving Fourier transform estimate and Littewood Paey theory, together with a decomosition of the sace Log L β )β>0). An interesting robem is that whether our resut can be imroved. As an easy coroary of Theorem, we have

3 G. Hu, D. Yan / J. Math. Ana. A ) Theorem. Let Ω be homogeneous of degree zero and have mean vaue zero on the unit shere and k be a ositive integer. If Ω Log L) k+ S n ), then for b BMOR n ),the commutator µ Ω;b,k is bounded on L R n ) with bound C b k BMOR n ) for a <<. Throughout this aer, C denotes the constants that are indeendent of the main arameters invoved but whose vaue may differ from ine to ine. For a measurabe set E, denote by χ E the characteristic function of E. Forf defined on R n, fˆ denotes the Fourier transform of f.. Proof of Theorem We begin with some reiminary emmas. Lemma see [4]). Let φ C0 R) be a radia function such that su φ {/4 ξ 4} and φ ξ ) =, ξ 0. Z Define the mutiier oerator S by Ŝ fξ)= φ ξ ) ˆ fξ), and S by S fx)= S S f )x). For any ositive integer k and b BMOR n ), denote by S ;b,k resectivey S;b,k ) the kth order commutator of S resectivey S ). Then for <<, i) Z S ;b,kf ) / n,k,) b k BMOR n ) f, ii) Z S ;b,k f ) / n,k,) b k BMOR n ) f. Lemma see [4]). Let 0 <δ<, m δ C0 Rn ) with suort contained in {δ/4 ξ 4δ}. Suose that for some ositive constant α, m δ min { δ α,δ α}, m δ. Let T δ be the mutiier oerator defined by T δ fξ)= m δ ξ) fξ). ˆ For a ositive integer k and b BMOR n ), denoted by T δ;b,k the kth order commutator of T δ. Then for any fixed 0 <ν<, there exists a ositive constant C = Cn,k,ν) such that T δ;b,k f min { δ αν,δ αν} b k BMOR n ) f. Lemma 3 see [4]). Let Ω be homogeneous of degree zero, k be a ositive integer and b BMOR n ). Suose that Ω beongs to the sace L S n ).Foreachs,set { λ Ω,s = inf λ>0: Ω og s + ) } Ω. λ λ

4 354 G. Hu, D. Yan / J. Math. Ana. A ) Then the oerator M Ω;b,k fx)= su r n r>0 x y <r bx) by) k Ωx y)fy) dy is bounded on L R n ) with bound Cλ Ω,k b k BMOR n ) for a <<. Lemma 4 see [4]). Let k be a ositive integer and b BMOR n ), Ω be homogeneous of degree zero and beong to L S n ).Forj Z,etσ j x) = x n Ωx)χ { j < x j+ } x). Denote by U j the convoution oerator whose kerne is σ j, and U j;b,k the kth order commutator of U j. Then the estimate ) / U j;b,k f j ) / λ Ω,k b k BMOR n ) f j 4) hods for any <<. Proof of Theorem. Without oss of generaity, we may assume that b BMOR n ) =. For each t>0andj Z,etK j,t x) = j t) Ωx) χ x n { x j t} x). Define the oerator F j,t by F j,t f )x) = Ωx y) j fy)dy. t x y n x y j t Denote by F j,t;b,k the kth order commutator of F j,t.sete 0 ={θ S n : Ωθ) } and E d ={θ S n : d < Ωθ) d+ } for ositive integer d. LetΩ d be the restriction of Ω on E d,i.e.,ω d θ) = Ωθ)χ Ed θ). Obviousy, Ω Log L) β S n ) is equivaent to that d= d β Ω d <.LetK j,d,t x) = j t) Ω d x) χ x n { x j t} x). Denote by F j,d,t the convoution oerator whose kerne is K j,d,t,andf j,d,t;b,k the kth order commutator of F j,d,t.lets be the mutiier oerator defined in Lemma. Write µ Ω;b,k f )x) = = = j+ j= j With the aid of the formua ) k k bx) by) = m=0 Ft;b,k f )x) t 3 F j,t;b,k f )x) t ) / ) / F j,t;b,k S j f ) ) x) /. 5) t Z C m k bx) bz) ) k m bz) by) ) m, x,y,z R n,

5 G. Hu, D. Yan / J. Math. Ana. A ) the Fubini theorem and a trivia comutation gives that F j,t;b,k S j f ) x) = k F j,t S j )b,k fx) Ck m F j,t;b,k m S j;b,m f ) x). On the other hand, we see that if m>0andf,h C0 Rn ),then hx)s j;b,m fx)dx Z R n m = Cm u hx) bx) mb b) ) u S j mb b) b ) m u ) f x) dx u=0 Z R n m = Cm u hx) bx) mb b) ) u mb b) bx) ) m u fx)dx= 0, u=0 Z R n where B is a ba with arge radius such that suf and sug are both contained in B, and m B b) is the mean vaue of b on B. Thus in this case, S j;b,m fx)= 0, a.e. x Rn. Z Therefore, by the Minkowski inequaity, µ Ω;b,k f )x) 0 = = m= F j,t S j ) b,k fx) ) / F j,0,t S j ) b,k fx) d= =Nd+ d= = ) / F j,d,t S j ) b,k fx) Nd Fj,d,t S ) j b,k fx) = Ifx)+ Jfx)+ Ufx)+ Vfx), where N is a ositive integer which wi be chosen ater. We consider the term I first. Let I fx)= / F j,t S j ) b,k fx) ). ) / ) /

6 356 G. Hu, D. Yan / J. Math. Ana. A ) Observe that I f = F j,t S j ) b,k f. Set m j,t ξ) = K j,t ξ), m j,t ξ) = m j,tξ)φ j ξ), and define the mutiier oerator Fj,t by F j,t f )ξ) = m j,t ξ) fξ). ˆ Straightforward comutation eads to that for t [, ] m j,t j ), m j,t j ), su m j,t j ξ ) { ξ +}. 6) Let F j,t be the oerator defined by F j,t f )ξ) = m j,t j ξ ) ˆ fξ). The Fourier transform estimate 6) via Lemma states us that for any 0 <ν<and nonnegative integer m, F j,t;b,m f ) Ω ν f, 0, t [, ]. By diation-invariance, we obtain F j,t;b,m f ) Ω ν f, 0, t [, ]. 7) Observe that for f,h C0 Rn ), k hx) S j F j,t )b,k fx)dx= R n m=0 It foows from the estimate 7) that C k m R n hx)f j,t;b,m S j;b,k m f )x) dx. S j F j,t )b,k f k Fj,t;b,m S j;b,k mf) which is equivaent to that m=0 ν Ω k S j;b,k m f m=0 ν Ω f, 0, t [, ], I f Ω ν f. 8) Aying the Minkowski inequaity and Lemma 4, we have that for <<,

7 I f G. Hu, D. Yan / J. Math. Ana. A ) F j,t S j R n k m=0 k m=0 k m=0 d 0 )b,k fx) ) / dx) / F j,t;b,m S j;b,k m f ) ) / F j,t,d;b,m S j;b,k m f ) ) / ) d 0 d 0 λ Ωd,k λ Ωd,k S j,k m f ) / ) ) f. 9) To estabish the L R n ) boundedness of I for the case of <<, we consider the maing F defined by F : { h j x) } { F j,t;b,m h j )x) }. By Lemma 3, we see that for each j Z, t [, ] and <<, F j,t;b,m h) F j,d,t;b,m h) λ Ωd,m h. Thus, for any < 0 <, Fj,t;b,m h j )x) 0 ) /0 0 dx λ Ωd,m) h j 0 0, R n 0 which tes us that the maing F is bounded from the sace L 0R n ) 0) to the sace L 0R n )L 0[, ]) 0)) with bound C d 0 λ Ω d,m. On the other hand, note that su su t [,] Fj,d,t;b,m h j )x) su su Fj,t;b,m h j )x) t [,] M Ωd ;b,m su h j ) x). Thus, for any < <, it foows from Lemma 3 that su su Fj,t;b,m h j ) t [,] λ Ωd,m su h j. This shows that the maing F is bounded from L R n ) ) to L R n ) L [, ]) )) with bound C d 0 λ Ω d,m. For each fixed with <<, we can

8 358 G. Hu, D. Yan / J. Math. Ana. A ) choose < 0 <, < < such that / = + 0 )/. By a standard interoation argument, we see that F is bounded from L R n ) ) to L R n )L [, ]) )) with bound C d 0 λ Ω d,m, thatis, Fj,t;b,m h j ) / ) d 0 Therefore, for <<, I f k m=0 d 0 k m=0 d 0 λ Ωd,m λ Ωd,m λ Ωd,m ) / h j, <<. Fj,t;b,m S j;b,k m f) / ) S j;b,k m f) ) / d 0 λ Ωd,k f. 0) Now we turn our attention to the terms J and U. For ositive integer, et and J fx)= U d fx)= F j,0,t S j ) b,k fx) ) / F j,d,t S j ) b,k fx) / ). By a we-known Fourier transform estimate of Duoandikoetxea and Rubio de Francia see [,. 55]), it is easy to show that if t [, ],then K j,d,t ξ) Ωd j ξ α, where α is a ositive constant deending ony on n. A trivia comutation gives that K j,d,t j Ω d, t [, ]. Define the oerator F j,d,t by F j,d,t f )ξ) = K j,d,t ξ)φ j ξ ) ˆ fξ). Lemma via diation-invariance says that for each nonnegative integer m, F j,d,t;b,m f ) α Ω d f. Consequenty,

9 G. Hu, D. Yan / J. Math. Ana. A ) U d f Simiary, we have k m=0 F j,d,t;b,m S j;b,k m f) α Ω d k S j;b,k m f m=0 α Ω d f, 0. ) J f α f, 0. ) By the same argument as that used in the roof of the inequaities 9) and 0), we can verify that U d f λ Ωd,k f, J f f, <<. 3) To estimate the term V, et Nd V d fx)= Fj,d,t S ) / j b,k fx) ). Observe that Nd V d fx) = k = m=0 Fj,d,t;b,k m S j;b,m f ) x) / ). As in the roof of the inequaities 9) and 0), we have that for <<, Write V d f Ndλ Ωd,k f. 4) V d f k m=0 Nd S j;b,k m S j F j,d,t ) b,m f ). = A standard duaity argument eads to that Nd S j;b,k m h = su = g C 0 Rn ), g = su g C 0 Rn ), g su g R n Nd Nd gx) S j;b,k m h x) dx R n = Nd h x)s j;b,k m gx)dx R n ) k m = h x) ) / Nd S j;b,k m gx) ) / dx = =

10 360 G. Hu, D. Yan / J. Math. Ana. A ) su g Nd Nd ) / Nd ) / h S j;b,k m g = = ) / h. This together with Lemma 4 in turn imies that k V d f Nd S j F j,d,t ) b,m f ) / m=0 = = k m Nd m=0 u=0 = k m Nd λ Ω d,u m=0 u=0 = Fj,d,t;b,u S j;b,m u f) S j;b,m u f Ndλ Ω d,k f. 5) We can now rove the L R n ) boundedness of µ Ω;b,k. Choose N>/α, whereα is the constant aeared in the inequaity ). Combining the estimates 8), ), ) and 5) yieds Note that µω;b,k f ) 0 = Ω d d k ogk Ω d + d I f + J f + = Ω + + d= =Nd+ d= =Nd+ α Ω d + ) Ω + + d / λ Ωd,k f. + d= ) Ω d d k Ω + d. d= U d f + V d f d= ) d / λ Ωd k f It is easy to verify that λ Ωd,k d k Ω d + d) and that µ Ω;b,k f ) Ω + + d /+k Ω d + d / d Ω d ) f. Therefore, Ω Log L) /+k S n ) is a sufficient condition such that µ Ω;b,k is bounded on L R n ). d= d=

11 G. Hu, D. Yan / J. Math. Ana. A ) It remains to rove the L R n ) boundedness of µ Ω;b,k for. We ony consider the case of <<. The case << can be roved by the same argument. Obviousy, C Ω d λ Ωd,k Ω d. Interoation the inequaities 8) and 0) gives that I f δ λ Ωd,k f, 0, 6) where δ = δ > 0. Simiary, it foows from the estimate ), ) and 3) that U d f γ Ω d f, J f γ f, >0, 7) where γ = γα,)>0. By the estimates 4) and 5), we can obtain that for any ε>0, V d f λ Ωd,kNd) /+ε f. 8) For given Ω Log L) k+β S n ) / <β<) and /β < <, take N>/γ)and ε>0 such that / + ε<β. Then we get that µ Ω;b,k f I f + J f + =0 = d= =Nd+ U d f + V d f d= ) λ Ωd,k + + d /+ε λ Ωd,k f f. This finishes the roof of Theorem. d= Acknowedgment The authors woud ike to thank the referee for some vauabe suggestions and corrections. References [] A.P. Caderón, A. Zygmund, On singuar integras, Amer. J. Math ) [] J. Duoandikoetxea, J.L. Rubio de Francia, Maxima and singuar integra oerators via Fourier transform estimates, Invent. Math ) [3] G. Hu, Moification of rough oerators, Ph.D. Thesis, Hangzhou University, Hangzhou, 993. [4] G. Hu, L R n ) boundedness for the commutators of homogeneous singuar integra oerators, Studia Math ) 3 7. [5] E.M. Stein, On the functions of Littewood Paey, Lusin, and Marcinkiewicz, Trans. Amer. Math. Soc ) [6] E.M. Stein, Harmonic Anaysis: Rea-Variabe Methods, Orthogonaity and Osciatory Integras, Princeton Univ. Press, Princeton, NJ, 993. [7] A. Torchinsky, S. Wang, A note on the Marcinkiewicz integra, Cooq. Math ) [8] T. Wash, On the function of Marcinkiewicz, Studia Math ) 03 7.