Kragujevac Journal of Matheatics Volue 34 200), Pages 03 2. SHARP FUNCTION ESTIMATE FOR MULTILINEAR COMMUTATOR OF LITTLEWOOD-PALEY OPERATOR PENG MEIJUN AND LIU LANZHE 2 Abstract. In this paper, we prove the sharp inequality for ultilinear coutator related to Littlewood-Paley operator. By using the sharp inequality, we obtain the weighted L p -nor inequality for the ultilinear coutator.. Introduction As the developent of singular integral operators, their coutators have been well studied see []-[4]). Let T be the Calderón-Zygund singular integral operator, a classical result of Coifan, Rocherberg and Weiss see [3]) states that coutator [b, T ]f) T bf) bt f) where b BMOR n )) is bounded on L p R n ) for < p <. In [7]-[9], the sharp estiates for soe ultilinear coutators of the Calderón- Zygund singular integral operators are obtained. The ain purpose of this paper is to prove the sharp inequality for ultilinear coutator related to the Littlewood- Paley operator. By using the sharp inequality, we obtain the weighted L p, L q )-nor inequality for the ultilinear coutator. 2. Notations and Results First let us introduce soe notations see [4], [9], [0]). In this paper, will denote a cube of R n with sides parallel to the axes, and for a cue let f fx)dx Key words and phrases. Multilinear coutator; Littlewood-Paley operator; BMO; Sharp inequality. 200 Matheatics Subject Classification. Priary: 42B20, Secondary: 42B25. Received: May 09, 2008 Revised: February 28, 200. 03
04 PENG MEIJUN AND LIU LANZHE 2 and the sharp function of f is defined by f # x) sup fy) f. x It is well-known that see [4]) f # x) sup inf x c C fy) C. We say that b belongs to BMOR n ) if b # belongs to L R n ) and define b BMO b # L. It has been known that see [0]) b b 2 k BMO Ck b BMO. Let M be the Har-Littlewood axial operator, that is that Mf)x) sup fy) ; x we write that M p f) M f p )) /p for 0 < p <. Let 0 < δ < n, 0 < r <, set /r M r,δ f)x) sup fy) ) r. x δr/n If 0 < r p < n/δ, /q /p δ/n, we know M r,δ is type of p, q), that is For b j BMOR n )j,..., ), set M r,δ f) q C f p. b BMO b j BMO. Given a positive integer and j, we denote by Cj the faily of all finite subsets σ {σ),..., σj)} of {,..., } of j different eleents. For σ Cj, set σ c {,..., } \ σ. For b b,..., b ) and σ {σ),..., σj)} C j, set bσ b σ),..., b σj) ), b σ b σ)... b σj) and b σ BMO b σ) BMO... b σj) BMO. In this paper, we will stu soe ultilinear coutators as following. Definition 2.. Suppose b j j,..., ) are the fixed locally integrable functions on R n. Let 0 < δ < n, ε > 0 and ψ be a fixed function which satisfies the following properties: ) Rn ψx)dx 0, 2) ψx) C + x ) n+ δ), 3) ψx + y) ψx) C y ε + x ) n++ε δ) when 2 y < x.
SHARP FUNCTION ESTIMATE FOR MULTILINEAR COMMUTATOR 05 The Littlewood-Paley ultilinear coutator is defined by where g b ψ,δ f)x) 0 F b t f)x) 2 dt ) /2, t F b t f)x) b j x) b j y)) ψ t x y)fy) R n and ψ t x) t n+δ ψx/t) for t > 0. Set F t f)x) R n ψ tx y)fy), we also define that g ψ,δ f)x) which is the Littlewood-Paley g function see []). 0 F t f)x) 2 dt ) /2, t Let H be the space H {h : h 0 ht) 2 dt/t) /2 }, then, for each fixed x R n, F b t f)x) ay be viewed as a apping fro [0, + ) to H, and it is clear that and g ψ,δ f)x) F t f)x) g b ψ,δ f)x) F b t f)x). Note that when b... b, g b ψ,δ is just the order coutator see [],[6]). In [5], the sharp estiates for the ultilinear coutator g b µ of another Littlewood-Paley operator g µ are obtained. It is well known that coutators are of great interest in haronic analysis and have been widely studied by any authors see []-[3], [6]-[9]). Our ain purpose is to establish the sharp inequality for the ultilinear coutator. Now we state our theores as following. Theore 2.. Let b j BMOR n ) for j,...,. Then for any < r <, there exists a constant C > 0 such that for any f C0 R n ) and any x R n, g b ψ,δ f)) # x) C M r,δ f)x) + M r g b σ c ψ,δ f))x). σ Cj Theore 2.2. Let b j BMOR n ) for j,...,. Then g b ψ,δ is bounded fro L p R n ) to L q R n ), where < p < n/δ, /q /p δ/n.
06 PENG MEIJUN AND LIU LANZHE 2 3. Proofs of Theores To prove the theores, we need the following lea. Lea 3. see []). Let 0 < δ < n, < p < n/δ, /q /p δ/n. Then g ψ,δ is bounded fro L p R n ) to L q R n ). Lea 3.2. and Let < r <, b j BMOR n ) for j,..., k. Then b j y) b j ) C b j BMO /r b j y) b j ) r C b j BMO. Proof. Choose < p j <, j,..., such that /p +... + /p, we obtain, by the Hölder s inequality, /pj b j y) b j ) b j y) b j ) p j ) C b j BMO and /r b j y) b j ) r C b j BMO. The lea follows. ) /pj r b j y) b j ) pjr Proof of Theore 2.. It suffices to prove for f C 0 R n ) and soe constant C 0, the following inequality holds: g b ψ,δ f)x) C 0 dx C b BMO M r,δ f) x) + Fix a cube x 0, d) and x. σ Cj M r g b σ c ψ,δ f) x)). We first consider the Case. Write, for f fχ 2 and f 2 fχ 2) c, F b t f)x) b x) b ) 2 )F t f)x) F t b b ) 2 )f )x) F t b b ) 2 )f 2 )x).
SHARP FUNCTION ESTIMATE FOR MULTILINEAR COMMUTATOR 07 Then, g b ψ,δ f)x) g ψ,δb ) 2 b )f 2 )x 0 ) F b t f)x) F t b ) 2 b )f 2 )x 0 ) F b t f)x) F t b ) 2 b )f 2 )x 0 ) b x) b ) 2 )F t f)x) + F t b b ) 2 )f )x) + F t b b ) 2 )f 2 )x) F t b b ) 2 )f 2 )x 0 ) Ax) + Bx) + Cx). For Ax), by the Hölder s inequality with exponent /r + /r, we get Ax)dx b x) b ) 2 g ψ,δ f)x) dx ) /r C b x) b ) 2 r dx g ψ,δ f)x) r dx 2 2 C b BMO M r g ψ,δ f)) x). For Bx), choose p such that < r < p < q < n/δ, /q /p δ/n, r ps, by the boundedness of g ψ,δ on L p R n )to L q R n ) and the Hölder s inequality, we obtain Bx)dx [g ψ,δ b b ) 2 )f )x)]dx ) /q [g ψ,δ b b ) 2 )fχ 2 )x)] q dx R n C ) /p b q x) b ) 2 p fx)χ 2 x) p dx R n C /q+/ps + δps/n)/ps ) /ps b x) b ) 2 ps dx 2 2 ) /ps fx) ps dx 2 δps/n 2 C /q+/ps + δr/n)/r ) /ps b x) b ) 2 ps dx 2 2 ) /r 2 δr/n fx) r dx C b BMO M r,δ f) x). ) /r
08 PENG MEIJUN AND LIU LANZHE 2 For Cx), by the Minkowski s inequality, we obtain Cx) F t b b ) 2 )f 2 )x) F t b b ) 2 )f 2 )x 0 ) ) 2 dt b y) b ) 2 fy) ψ t x y) ψ t x 0 y) 0 2) c t b y) b ) 2 fy) 2) c 0 C b y) b ) 2 fy) 2) c 0 x 0 x ε C b y) b ) 2 fy) 2) c x 0 y n+ε δ x 0 x ε C b y) b ) 2 fy) k 2 k+ \2 k x 0 y ) /r C 2 kε fy) r k 2 k+ δr/n 2 k+ ) r b 2 k+ y) b ) 2 r 2 k+ C k2 kε b BMO M r,δ f) x) k C b BMO M r,δ f) x), ) /2 t ψ tx y) ψ t x 0 y) 2 dt x 0 x 2ε tdt t + x 0 y ) 2n++ε δ) n+ε δ ) /2 /2 thus Cx)dx C b BMO M r,δ f) x). Now, we consider the Case 2, we have known that, for b b,..., b ), F b t f)x) b j x) b j y)) ψ t x y)fy) R n R n b j x) b j ) 2 ) b j y) b j ) 2 )ψ t x y)fy) ) j bx) b) 2 ) σ by) b) 2 ) σ ψ t x y)fy) j0 σ C R j n
SHARP FUNCTION ESTIMATE FOR MULTILINEAR COMMUTATOR 09 b x) b ) 2 )... b x) b ) 2 )F t f)x) + ) F t b b ) 2 )... b b ) 2 )f)x) + σ Cj ) j bx) b) 2 ) σ R n by) bx)) σ cψ t x y)fy) b x) b ) 2 )... b x) b ) 2 )F t f)x) + ) F t b b ) 2 )... b b ) 2 )f)x) + σ Cj c,j bx) b) 2 ) σ F b σ c t f)x), thus, + g b ψ,δ f)x) g ψ,δ b ) 2 b )... b ) 2 b ))f 2 )x 0 ) F b t f)x) F t b ) 2 b )... b )2 b )f 2 )x 0 ) F b t f)x) F t b ) 2 b )... b ) 2 b )f 2 )x 0 ) b x) b ) 2 )... b x) b ) 2 )F t f)x) σ Cj bx) b ) 2 ) σ F b σ c t f)x) + F t b b ) 2 )... b b ) 2 )f )x) + F t b j b j ) 2 )f 2 )x) F t b j b j ) 2 )f 2 )x 0 ) I x) + I 2 x) + I 3 x) + I 4 x). For I x), by the Hölder s inequality with exponent /p +... + /p + /r, where < p j <, j,...,, we get I x)dx b x) b ) 2... b x) b ) 2 g ψ,δ f)x) dx b x) b ) 2 p ) /p... g ψ,δ f)x) r dx ) /r ) /p b x) b ) 2 p dx C b BMO M r g ψ,δ f)) x).
0 PENG MEIJUN AND LIU LANZHE 2 For I 2 x), by the Minkowski s and Hölder s inequality, we get I 2 x)dx σ Cj C C σ Cj σ Cj σ Cj bx) b) 2 ) σ F b σ c t f)x) dx bx) b) 2 ) σ g b σ c ψ,δ f)x) dx ) /r ) /r bx) b) 2 ) σ r dx g b σ c ψ,δ 2 2 f)x) r dx b σ BMO M r g b σ c ψ,δ f)) x). For I 3 x), choose < r < p < q < n/δ, /q /p δ/n, r ps, by the boundedness of g ψ,δ fro L p R n ) to L q R n ), and Hölder s inequality, we get I 3 x)dx F t b j b j ) 2 )f )x) dx /q g ψ,δ b j b j ) 2 )fχ 2 )x) q dx R n C /p b /q j x) b j ) 2 ) p fx)χ 2 x) p dx R n C /ps ) /ps b /q j x) b j ) 2 ) ps dx fx) ps dx 2 2 C /q+/ps δps/n)/ps) /ps b j y) b j ) 2 ) ps dx 2 2 ) /ps fx) ps dx 2 δps/n 2 C b BMO M r,δ f) x). For I 4 x), choose < p j <, j,..., such that /p +... + /p + /r, we obtain, by the Hölder s inequality,
SHARP FUNCTION ESTIMATE FOR MULTILINEAR COMMUTATOR I 4 x) F t b j b j ) 2 )f 2 )x) F t b j b j ) 2 )f 2 )x 0 ) 2 /2 b j y) b j ) 2 ) fχ 2) cy)ψ t x y) ψ t x 0 y)) dt 0 R n t ψ t x y) ψ t x 0 y) 2 /2 dt C b j y) bb j ) 2 ) fχ 2) cy) dt) R n 0 t x x 0 2ε ) /2 tdt C b j y) b j ) 2 ) fy) 2) c 0 t + x 0 y ) 2n++ε δ) x x 0 ε C b j y) b j ) 2 ) fy) 2) c x 0 y n+ε δ) C x x 0 ε x 0 y n+ε δ) b j y) b j ) 2 ) f 2 y) k 2 k+ \2 k C 2 δ kε 2 k+ +rδ/n b j y) b j ) 2 ) f 2 y) k 2 k+ ) /r C 2 kε fy) r k 2 k+ δr/n 2 k+ ) /pj b 2 k+ j y) b ) 2 p j 2 k+ C k 2 k b j BMO M r,δ f) x) k C b BMO M r,δ f) x), thus I 4 x)dx C b BMO M r,δ f) x). This copletes the proof of the theore.
2 PENG MEIJUN AND LIU LANZHE 2 Proof of Theore 2.2. We first consider the case, we have g b ψ,δ f) L q Mgb ψ,δ )f) L q C gb ψ,δ f))# L q C M r g ψ,δ f)) L q + C M r,δ f) L q C g ψ,δ f) L q + C M r,δ f) L q C f L p + C f L p C f L p. When 2, we ay get the conclusion of the theore by induction. This finishes the proof. References [] J. Alvarez, R. J. Babgy, D. S. Kurtz and C. Pérez, Weighted estiates for coutators of linear operators, Studia Math. 04 993), 95 209. [2] R. Coifan and Y. Meyer, Wavelets, Caldrón-Zygund and ultilinear operarors, Cabridge studies in Advanced Math. 48, Caridge University Press, Cabridge, 997. [3] R. Coifan, R. Rochberg and G. Weiss, Factorization theores for Har spaces in several variables, Ann. of Math. 03 976), 6 635. [4] J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted nor inequalities and related topics, North-Holland Math. 6, Asterda, 985. [5] J. L. Hao, L. Z. Liu, Sharp estiates for ultilinear coutator of Littlewood - Paley operator, Coun. Korean Math. Soc., 23 ) 2008), 49-59. [6] L. Z. Liu, Weighted weak type estiates for coutators of Littlewood-Paley operator, Japanese J. of Math. 29 2003), 3. [7] C. Pérez, Endpoint estiate for coutators of singular integral operators, J. Func. Anal. 28 995), 63 85. [8] C. Pérez and G. Pradolini, Sharp weighted endpoint estiates for coutators of singular integral operators, Michigan Math. J. 49 200), 23 37. [9] C. Pérez and R. Trujillo-Gonzalez, Sharp Weighted estiates for ultilinear coutators, J. London Matha. Soc. 65 2002), 672 692. [0] E. M. Stein, Haronic analysis: real variable ethods, orthogonality and oscillatory integrals, Princeton Univ. Press, Princeton NJ, 993. [] A. Torchinsky, Real variable ethods in haronic analysis, Pure and Applied Math., 23, Acadeic Press, New York, 986. Departent of Matheatics, Chang Jun Fu Rong Middle School, Ma Wang Dui Zhong Road, Changsha, Hunan Province, 4000, P. R. of China E-ail address: pengeijun00@63.co 2 Departent of Matheatics, Changsha university of Science and Technology, Changsha, Hunan Province, 40077, P. R. of China E-ail address: lanzheliu@63.co