Sharp Maximal Function Estimates and Boundedness for Commutator Related to Generalized Fractional Integral Operator

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1 DOI: /v z Analele Universităţii de Vest, Timişoara Seria Matematică Informatică L, 2, 202), 97 5 Sharp Maximal Function Estimates Boundedness for Commutator Related to Generalized Fractional Integral Operator Guo Sheng, Huang Chuangxia, Liu Lanzhe Astract. In this paper, we prove some sharp maximal function estimates for the commutators related to certain generalized fractional integral operator with general kernel the BM O Lipschitz functions. As an application, we otain the oundedness of the commutators on Leesgue, Morrey Trieel-Lizorkin spaces. The operator includes fractional Littlewood-Paley operators, Marcinkiewicz operators Bochner-Riesz operator. AMS Suject Classification 2000). 42B20, 42B25. Keywords. Commutator; Littlewood-Paley operator; Marcinkiewicz operator; Bochner-Riesz operator; Sharp maximal function; Morrey space; Trieel-Lizorkin space; BM O; Lipschitz function. Introduction As the development of singular integral operatorssee [9][24][25]), their commutators have een well studiedsee [6][22][23]). In [8][22][23], the authors prove that the commutators generated y the singular integral operators BMO functions are ounded on L p R n ) for < p <. Chanillo see [2]) proves a similar result when singular integral operators are replaced y the Authenticated Download Date 8//3 9:46 AM

2 98 G. Sheng H. Chuangxia L. Lanzhe An. U.V.T. fractional integral operators. In [3][0][9], the oundedness for the commutators generated y the singular integral operators Lipschitz functions on Trieel-Lizorkin L p R n ) < p < ) spaces are otained. In [], some singular integral operators with general kernel are introduced, the oundedness for the operators their commutators generated y BM O Lipschitz functions are otainedsee [][]). In this paper, certain generalized fractional integral operators with general kernel are introduced, the sharp maximal function estimates for the commutator related to the operator the BM O Lipschitz functions are otained, as an application, we otain the oundedness of the commutator on Leesgue, Morrey Trieel-Lizorkin space. The operator includes fractional Littlewood-Paley operators, Marcinkiewicz operators Bochner-Riesz operator. 2 Preliminaries First, let us introduce some notations. Throughout this paper, will denote a cue of R n with sides parallel to the axes. For any locally integrale function f, the sharp maximal function of f is defined y M # f)x) = sup fy) f dy, x where, in what follows, f = fx)dx. It is well-known that see [9][24]) M # f)x) sup inf fy) c dy. x c C We say that f elongs to BMOR n ) if M # f) elongs to L R n ) define f BMO = M # f) L. It has een known that see [20]) Let f f 2 k BMO Ck f BMO. Mf)x) = sup fy) dy. x For η > 0, let M η f)x) = M f η ) /η x). For 0 < η < n r <, set M η,r f)x) = sup x rη/n fy) r dy) /r. Authenticated Download Date 8//3 9:46 AM

3 Vol. L 202) Sharp Maximal Function Estimates Boundedness for Commutator...99 The A p weight is defined y see [9]) { A p = w L locr n ) : sup for < p < ) p wx)dx wx) dx) /p ) < A = {w L p loc Rn ) : Mw)x) Cwx), a.e.}. For β > 0 p >, let F p β, R n ) e the homogeneous Trieel-Lizorkin spacesee [9]). For β > 0, the Lipschitz space Lip β R n ) is the space of functions f such that f Lipβ = sup x,y R n x y fx) fy) x y β <. In this paper, we will study some integral operators as followingsee []). Definition. Fixed 0 δ < n. Let F δ,t x, y) e the function defined on R n R n [0, + ) e a locally integrale function on R n, set F δ,t f)x) = F δ,t x, y)fy)dy R n F δ,t f)x) = x) y))f δ,t x, y)fy)dy R n for every ounded compactly supported function f. Let H e the Banach space H = {h : h < }. For each fixed x R n, we view F δ,t f)x) F δ,t f)x) as the mappings from [0, + ) to H. Set T δ f)x) = F δ,t f)x), which T δ is ounded on L 2 R n ). The commutator related to F δ,t is defined y T δ f)x) = F δ,t f)x) F δ,t satisfies: there is a sequence of positive constant numers {C k } such that for any k, F δ,t x, y) F δ,t x, z) + F δ,t y, x) F δ,t z, x) )dx C, 2 y z < x y F δ,t x, y) F δ,t x, z) + F δ,t y, x) F δ,t z, x) ) q dy 2 k z y x y <2 k+ z y }, ) /q Authenticated Download Date 8//3 9:46 AM

4 00 G. Sheng H. Chuangxia L. Lanzhe An. U.V.T. C k 2 k z y ) n/q +δ, where < q < 2 /q + /q =. Definition 2. Let ϕ e a positive, increasing function on R + there exists a constant D > 0 such that ϕ2t) Dϕt) for t 0. Let f e a locally integrale function on R n. Set, for p <, f L p,ϕ = /p sup fy) dy) p, x R n, d>0 ϕd) x,d) where x, d) = {y R n : x y < d}. The generalized Morrey space is defined y L p,ϕ R n ) = {f L locr n ) : f L p,ϕ < }. If ϕd) = d δ, δ > 0, then L p,ϕ R n ) = L p,δ R n ), which is the classical Morrey spaces see [20][2]). If ϕd) =, then L p,ϕ R n, w) = L p R n ), which is the Leesgue spaces. As the Morrey space may e considered as an extension of the Leesgue space, it is natural important to study the oundedness of the operator on the Morrey spaces see [4][7][8][2][7]). It is well known that commutators are of great interest in harmonic analysis have een widely studied y many authors see [22][23]). In [23], Pérez Trujillo-Gonzalez prove a sharp estimate for the commutator. The main purpose of this paper is to prove the sharp maximal inequalities for the commutator T δ. As the application, we otain the Lp -norm inequality, Morrey Trieel-Lizorkin spaces oundedness for the commutator. 3 Theorems Lemmas We shall prove the following theorems. Theorem. Let T δ e the integral operator as Definition, the sequence {C k } l, 0 < β <, q s < Lip β R n ). Then there exists a constant C > 0 such that, for any f C 0 R n ) x R n, M # T δ f)) x) C Lipβ Mβ+δ,s f) x) + M β,s T δ f)) x) ). Authenticated Download Date 8//3 9:46 AM

5 Vol. L 202) Sharp Maximal Function Estimates Boundedness for Commutator...0 Theorem 2. Let T δ e the integral operator as Definition, the sequence {2 kβ C k } l, 0 < β <, q s < Lip β R n ). Then there exists a constant C > 0 such that, for any f C0 R n ) x R n, sup inf x c +β/n T δ f)x) c dx C Lipβ Mδ,s f) x) + M s T δ f)) x) ). Theorem 3. Let T δ e the integral operator as Definition, the sequence {kc k } l, q s < BMOR n ). Then there exists a constant C > 0 such that, for any f C 0 R n ) x R n, M # T δ f)) x) C BMO Mδ,s f) x) + M s T δ f)) x) ). Theorem 4. Let T δ e the integral operator as Definition, the sequence {C k } l, 0 < β < min, n δ), q < p < n/β + δ), /r = /p β + δ)/n Lip β R n ). Then T δ is ounded from Lp R n ) to L r R n ). Theorem 5. Let T δ e the integral operator as Definition, the sequence {C k } l, 0 < D < 2 n, 0 < β < min, n δ), q < p < n/β + δ), /r = /p β + δ)/n Lip β R n ). Then T δ is ounded from Lp,ϕ R n ) to L r,ϕ R n ). Theorem 6. Let T δ e the integral operator as Definition, the sequence {2 kβ C k } l, 0 < β <, q < p < n/δ, /r = /p δ/n Lip β R n ). Then T δ is ounded from Lp R n ) to F r β, R n ). Theorem 7. Let T δ e the integral operator as Definition, the sequence {kc k } l, q < p < n/δ, /r = /p δ/n BMOR n ). Then T δ is ounded from L p R n ) to L r R n ). Theorem 8. Let T δ e the integral operator as Definition, the sequence {kc k } l, 0 < D < 2 n, q < p < n/δ, /r = /p δ/n BMOR n ). Then T δ is ounded from Lp,ϕ R n ) to L r,ϕ R n ). To prove the theorems, we need the following lemma. Lemma.see []) Let T δ e the integral operator as Definition. Then T δ is ounded from L p R n ) to L r R n ) for < p < n/δ /r = /p δ/n. Lemma 2.see [9]). For 0 < β < < p <, we have f F p β, sup sup inf c +β/n +β/n fx) f dx fx) c dx L p. L p Lemma 3.see [9]). Let 0 < p < w r< A r. Then, for any smooth function f for which the left-h side is finite, Mf)x) p wx)dx C M # f)x) p wx)dx. R n R n Authenticated Download Date 8//3 9:46 AM

6 02 G. Sheng H. Chuangxia L. Lanzhe An. U.V.T. Lemma 4.see [2]) Suppose that 0 < η < n, < s < p < n/η /q = /p η/n. Then M η,s f) L q C f L p. Lemma 5. Let < p <, 0 < D < 2 n. Then, for any smooth function f for which the left-h side is finite, Mf) L p,ϕ C M # f) L p,ϕ. Proof. For any cue = x 0, d) in R n, we know Mχ ) A for any cue = x, d) y [5]. Noticing that Mχ ) Mχ )x) d n / x x 0 d) n if x c, y Lemma 3, we have, for f L p,ϕ R n ), Mf)x) p dx = Mf)x) p χ x)dx R n Mf)x) p Mχ )x)dx R n C M # f)x) p Mχ )x)dx R n ) = C M # f)x) p Mχ )x)dx + M # f)x) p Mχ )x)dx k=0 2 k+ \2 k ) C M # f)x) p dx + M # f)x) p k=0 2 k+ \2 k 2 k+ dx ) C M # f)x) p dx + M # f)x) p 2 kn dy 2 k+ k=0 C M # f) p L 2 kn ϕ2 k+ d) p,ϕ k=0 C M # f) p L 2 n D) k ϕd) p,ϕ k=0 C M # f) p L p,ϕϕd), thus /p ) /p Mf)x) dx) p C M # f)x) p dx ϕd) ϕd) Mf) L p,ϕ C M # f) L p,ϕ. Authenticated Download Date 8//3 9:46 AM

7 Vol. L 202) Sharp Maximal Function Estimates Boundedness for Commutator...03 This finishes the proof. Lemma 6. Let T δ e the integral operator as Definition, 0 < D < 2 n, < p < n/δ /r = /p δ/n. Then T δ f) L r,ϕ C f L p,ϕ. Lemma 7. Let 0 < D < 2 n, s < p < n/η /r = /p η/n. Then M η,s f) L r,ϕ C f L p,ϕ. The proofs of two Lemmas are similar to that of Lemma 5 y Lemma 4, we omit the details. 4 Proofs of Theorems Proof of Theorem. It suffices to prove for f C0 R n ) some constant C 0, the following inequality holds: T δ f)x) C 0 dx C Lipβ Mβ+δ,s f) x) + M β,s T δ f)) x) ). Fix a cue = x 0, d) x. Write, for f = fχ 2 f 2 = fχ 2) c, F δ,t f)x) = x) 2 )F δ,t f)x) F δ,t 2 )f )x) F δ,t 2 )f 2 )x). Then + F δ,t f)x) F δ,t 2 )f 2 )x 0 ) dx x) 2 )F δ,t f)x) dx + F δ,t 2 )f )x) dx F δ,t 2 )f 2 )x) F δ,t 2 )f 2 )x 0 ) dx = I + I 2 + I 3. For I, y Hölder s inequality, we otain I sup x) 2 T δ f)x) dx x 2 sup x) 2 /s T δ f)x) s dx x 2 C Lipβ /s 2 β/n /s β/n C Lipβ M β,s T δ f)) x). sβ/n ) /s ) /s T δ f)x) s dx Authenticated Download Date 8//3 9:46 AM

8 04 G. Sheng H. Chuangxia L. Lanzhe An. U.V.T. For I 2, choose < r < such that /r = /s δ/n, y L s, L r )-oundedness of T δ see Lemma ), we get I 2 ) /r T δ 2 )f )x) r dx R n C /r R n x) 2 )f x) s dx ) /s C /r Lipβ 2 β/n 2 /s β+δ)/n C Lipβ M β+δ,s f) x). 2 sβ+δ)/n 2 ) /s fx) s dx For I 3, recalling that s > q, we have I 3 y) 2 fy) F δ,t x, y) F δ,t x 0, y) dydx 2) c F δ,t x, y) F δ,t x 0, y) k= + k= 2 k d y x 0 <2 k+ d 2 k d y x 0 <2 k+ d y) 2 k+ fy) dydx F δ,t x, y) F δ,t x 0, y) 2 k+ 2 fy) dydx C F δ,t x, y) F δ,t x 0, y) q dy k= 2 k d y x 0 <2 k+ d ) /q fy) q dy dx 2 k+ sup y) 2 k+ y 2 k+ + C k= 2 k+ 2 fy) q dy 2 k+ ) /q dx ) /q F δ,t x, y) F δ,t x 0, y) q dy 2 k d y x 0 <2 k+ d ) /q Authenticated Download Date 8//3 9:46 AM

9 Vol. L 202) Sharp Maximal Function Estimates Boundedness for Commutator...05 C C k 2 k d) n/q +δ 2 k+ β/n Lipβ 2 k+ /q /s 2 k+ /s β+δ)/n k= ) /s fy) s dy 2 k+ sβ+δ)/n 2 k+ +C Lipβ 2 k β/n C k 2 k d) n/q δ 2 k+ /q /s 2 k+ /s β+δ)/n k= 2 k+ sβ+δ)/n C Lipβ M β+δ,s f) x) C Lipβ M β+δ,s f) x). 2 k+ k= C k ) /s fy) s dy These complete the proof of Theorem. Proof of Theorem 2. It suffices to prove for f C0 R n ) some constant C 0, the following inequality holds: T δ +β/n f)x) C 0 dx C Lipβ Mδ,s f) x) + M s T δ f)) x) ). Fix a cue = x 0, d) x. Write, for f = fχ 2 f 2 = fχ 2) c, F δ,t f)x) = x) 2 )F δ,t f)x) F δ,t 2 )f )x) F δ,t 2 )f 2 )x). Then +β/n +β/n + + +β/n +β/n F δ,t f)x) F δ,t 2 )f 2 )x 0 ) dx x) 2 )F δ,t f)x) dx F δ,t 2 )f )x) dx F δ,t 2 )f 2 )x) F δ,t 2 )f 2 )x 0 ) dx = II + II 2 + II 3. By using the same argument as in the proof of Theorem, we get, for Authenticated Download Date 8//3 9:46 AM

10 06 G. Sheng H. Chuangxia L. Lanzhe An. U.V.T. < r < with /r = /s δ/n, II ) /s C sup x) +β/n 2 /s T δ f)x) s dx x 2 C β/n Lipβ 2 β/n /s /s C Lipβ M s T δ f)) x), ) /s T δ f)x) s dx II 2 /r T δ +β/n 2 )f )x) r dx R n C /r x) +β/n 2 )f x) s dx R n C +β/n /r Lipβ 2 β/n 2 /s δ/n C Lipβ M δ,s f) x), ) /r ) /s 2 sδ/n 2 ) /s fx) s dx II 3 +β/n +β/n + +β/n y) 2 fy) F δ,t x, y) F δ,t x 0, y) dydx 2) c F δ,t x, y) F δ,t x 0, y) k= k= 2 k d y x 0 <2 k+ d 2 k d y x 0 <2 k+ d y) 2 k+ fy) dydx F δ,t x, y) F δ,t x 0, y) 2 k+ 2 fy) dydx C +β/n F δ,t x, y) F δ,t x 0, y) q dy k= 2 k d y x 0 <2 k+ d ) /q fy) q dy dx 2 k+ sup y) 2 k+ y 2 k+ ) /q Authenticated Download Date 8//3 9:46 AM

11 Vol. L 202) Sharp Maximal Function Estimates Boundedness for Commutator C +β/n 2 k+ 2 k= F δ,t x, y) F δ,t x 0, y) q dy 2 k d y x 0 <2 k+ d ) /q fy) q dy dx 2 k+ ) /q C β/n C k 2 k d) n/q +δ 2 k+ β/n Lipβ 2 k+ /q δ/n k= 2 k+ sδ/n +C β/n k= 2 k+ sδ/n C Lipβ M δ,s f) x) C Lipβ M δ,s f) x). 2 k+ ) /s fy) s dy Lipβ 2 k β/n C k 2 k d) n/q +δ 2 k+ /q δ/n 2 k+ ) /s fx) s dx 2 kβ C k k= This completes the proof of Theorem 2. Proof of Theorem 3. It suffices to prove for f C0 R n ) some constant C 0, the following inequality holds: T δ f)x) C 0 dx C BMO Mδ,s f) x) + M s T δ f)) x) ). Fix a cue = x 0, d) x. Write, for f = fχ 2 f 2 = fχ 2) c, F δ,t f)x) = x) 2 )F δ,t f)x) F δ,t 2 )f )x) F δ,t 2 )f 2 )x). Then + f)x) F δ,t 2 )f 2 )x 0 ) dx x) 2 )F δ,t f)x) dx + F δ,t 2 )f )x) dx F δ,t 2 )f 2 )x) F δ,t 2 )f 2 )x 0 ) dx F δ,t = III + III 2 + III 3. Authenticated Download Date 8//3 9:46 AM

12 08 G. Sheng H. Chuangxia L. Lanzhe An. U.V.T. For III, y Hölder s inequality, we get III ) /s ) /s x) 2 s dx T δ f)x) s dx C BMO M s T δ f)) x). For III 2, choose < p < s such that /r = /p δ/n, y Hölder s inequality L p, L r )-oundedness of T δ, we otain III 2 ) /r T δ 2 )f )x) r dx R n C /r R n 2 )f x) p dx ) /p ) s p)/sp C /r x) 2 sp/s p) dx fx) s dx 2 2 C /r 2 s p)/sp x) 2 sp/s p) dx 2 2 ) /s 2 /s δ/n fx) s dx 2 sδ/n C BMO /r+/p δ/n C BMO M δ,s f) x). 2 2 sδ/n 2 ) /s fx) s dx ) s p)/sp ) /s For III 3, recalling that s > q, taking < p < with /p + /q + /s =, y Hölder s inequality, we otain III 3 k= k= 2 k d y x 0 <2 k+ d F δ,t x, y) F δ,t x 0, y) y) 2 fy) dydx F δ,t x, y) F δ,t x 0, y) q dy 2 k d y x 0 <2 k+ d ) /p x) 2 p dx 2 k+ /s fy) dy) s dx 2 k+ ) /q Authenticated Download Date 8//3 9:46 AM

13 Vol. L 202) Sharp Maximal Function Estimates Boundedness for Commutator...09 C C k 2 k d) n/q +δ k2 k d) n/p BMO 2 k d) n/s δ/n) k= 2 k+ sδ/n 2 k+ ) /s fy) s dy C BMO kc k 2 k d) n +/q+/p+/s) 2 k+ sδ/n k= C BMO M δ,s f) x) C BMO M δ,s f) x). kc k k= 2 k+ ) /s fy) s dy This completes the proof of Theorem 3. Proof of Theorem 4. Choose q < s < p in Theorem let /t = /p δ/n, then /r = /t β/n, thus, we have, y Lemma, 3 4, T δ f) L r MT δ f)) L r C M # T δ f)) L r ) C Lipβ Mβ,s T δ f)) L r + M β+δ,s f) L r C Lipβ T δ f) L t + f L p) C Lipβ f L p. This completes the proof of Theorem 4. Proof of Theorem 5. Choose q < s < p in Theorem let /t = /p δ/n, then /r = /t β/n, thus, we have, y Lemma 5-7, T δ f) L r,ϕ MT δ f)) L r,ϕ C M # T δ f)) L r,ϕ ) C Lipβ Mβ,s T δ f)) L r,ϕ + M β+δ,s f) L r,ϕ C Lipβ T δ f) L t,ϕ + f L p,ϕ) C Lipβ f L p,ϕ. This completes the proof of Theorem 5. Proof Theorem 6. Choose q < s < p in Theorem 2. By using Lemma 2, we otain T δ f) F β, r C sup T δ +β/n f)x) T δ 2 )f 2 )x 0 ) dx L ) r C Lipβ Ms T δ f)) L r + M δ,s f) L r C Lipβ T δ f) L r + f L p) C Lipβ f L p. Authenticated Download Date 8//3 9:46 AM

14 0 G. Sheng H. Chuangxia L. Lanzhe An. U.V.T. This completes the proof of Theorem 6. Proof of Theorem 7. Choose q s < p in Theorem 3 similar to the proof of Theorem 4, we have T δ f) L r MT δ f)) L r C M # T δ f)) L r ) C BMO Ms T δ f)) L r + M δ,s f) L r C BMO T δ f) L r + f L p) C BMO f L p. This completes the proof of the theorem. Proof of Theorem 8. Choose q s < p in Theorem 3 similar to the proof of Theorem 5, we have T δ f) L r,ϕ MT δ f)) L r,ϕ C M # T δ f)) L r,ϕ ) C BMO Ms T δ f)) L r,ϕ + M δ,s f) L r,ϕ C BMO T δ f) L r,ϕ + f L p,ϕ) C BMO f L p,ϕ. This completes the proof of the theorem. 5 Applications In this section we shall apply the Theorems -8 of the paper to some particular operators such as the Littlewood-Paley operators, Marcinkiewicz operator Bochner-Riesz operator. Application. Littlewood-Paley operators. Fixed 0 δ < n, ε > 0 µ > 3n + 2)/n. Let ψ e a fixed function which satisfies: ) ψx)dx = 0, R n 2) ψx) C + x ) n+ δ), 3) ψx + y) ψx) C y ε + x ) n++ε δ) when 2 y < x ; We denote Γx) = {y, t) R+ n+ : x y < t} the characteristic function of Γx) y χ Γx). The Littlewood-Paley commutators are defined y gψf)x) = 0 Ft f)x) 2 dt ) /2, t [ Sψf)x) = Ft f)x, y) 2 dydt Γx) t n+ ] /2 Authenticated Download Date 8//3 9:46 AM

15 Vol. L 202) Sharp Maximal Function Estimates Boundedness for Commutator... g µf)x) = [ R n+ + ) ] nµ t Ft f)x, y) 2 dydt /2, t + x y t n+ where Ft f)x) = x) y))ψ t x y)fy)dy, R n Ft f)x, y) = x) z))fz)ψ t y z)dz R n ψ t x) = t n ψx/t) for t > 0. Set F t f)y) = f ψ t y). We also define g µ f)x) = g ψ f)x) = 0 F t f)x) 2 dt ) /2, t S ψ f)x) = F t f)y) 2 dydt Γx) t n+ R n+ + ) /2 ) ) nµ t F t f)y) 2 dydt /2, t + x y t n+ which are the Littlewood-Paley operators see [25]). Let H e the space { } /2 H = h : h = ht) dt/t) 2 < 0 or H = h : h = R n+ + ) /2 hy, t) 2 dydt/t n+ <, then, for each fixed x R n, F t f)x) F t f)x, y) may e viewed as the mapping from [0, + ) to H, it is clear that g ψf)x) = F t f)x), g ψ f)x) = F t f)x), S ψf)x) = χ Γx) F t f)x, y), Sψ f)x) = χγx) F t f)y) ) nµ/2 gµf)x) t = F t f)x, y) t + x y, Authenticated Download Date 8//3 9:46 AM

16 2 G. Sheng H. Chuangxia L. Lanzhe An. U.V.T. ) nµ/2 t g µ f)x) = F t f)y) t + x y. It is easily to see that gψ, S ψ g µ satisfy the conditions of Theorems -8see [3-5]), thus theorems -8 hold for gψ, S ψ g µ. Application 2. Marcinkiewicz operators. Fixed 0 δ < n, λ > max, 2n/n + 2)) 0 < γ. Let Ω e homogeneous of degree zero on R n with Ωx )dσx ) = 0. Assume that S n Ω Lip γ S n ). The Marcinkiewicz commutators are defined y µ λf)x) = µ Ωf)x) = 0 Ft f)x) 2 dt ) /2, t 3 [ µ Sf)x) = Ft f)x, y) 2 dydt Γx) t n+3 [ R n+ + ] /2 ) ] nλ t Ft f)x, y) 2 dydt /2, t + x y t n+3 where Set We also define Ft Ωx y) f)x) = x) y)) fy)dy x y t x y n δ Ft Ωy z) f)x, y) = x) z)) fz)dz. y z t y z n δ µ λ f)x) = F t f)x) = x y t µ Ω f)x) = µ S f)x) = R n+ + 0 Ωx y) fy)dy. x y n δ F t f)x) 2 dt ) /2, t 3 F t f)y) 2 dydt Γx) t n+3 ) /2 ) ) nλ t F t f)y) 2 dydt /2, t + x y t n+3 Authenticated Download Date 8//3 9:46 AM

17 Vol. L 202) Sharp Maximal Function Estimates Boundedness for Commutator...3 which are the Marcinkiewicz operatorssee [26]). Let H e the space { ) } /2 H = h : h = ht) 2 dt/t 3 < or H = h : h = Then, it is clear that 0 R n+ + ) /2 hy, t) 2 dydt/t n+3 <. µ Ωf)x) = F t f)x), µ Sf)x) = χγx) F t f)x, y), µ Ω f)x) = F t f)x), µs f)x) = χγx) F t f)y) ) nλ/2 µ t λf)x) = F t f)x, y) t + x y, ) nλ/2 t µ λ f)x) = F t f)y) t + x y. It is easily to see that µ Ω, µ S µ λ satisfy the conditions of Theorems -8 see [3-5][26]), thus Theorems -8 hold for µ Ω, µ S µ λ. Application 3. Bochner-Riesz operator. Let δ > n )/2, Bt δ f ˆ)ξ) = t 2 ξ 2 ) δ ˆfξ) + Bt δ z) = t n B δ z/t) for t > 0. Set Fδ,tf)x) = x) y))bt δ x y)fy)dy, R n The maximal Bochner-Riesz commutator is defined y We also define that Bδ, f)x) = sup Bδ,tf)x). t>0 B δ, f)x) = sup Bt δ f)x) t>0 which is the maximal Bochner-Riesz operatorsee [6]). Let H e the space H = {h : h = sup ht) < }, then t>0 B δ, f)x) = B δ,tf)x), B δ f)x) = B δ t f)x). It is easily to see that Bδ, satisfies the conditions of Theorems -8 with δ = 0see [3]), thus Theorems -8 hold for Bδ,. Authenticated Download Date 8//3 9:46 AM

18 4 G. Sheng H. Chuangxia L. Lanzhe An. U.V.T. References [] D. C. Chang, J. F. Li, J. Xiao, Weighted scale estimates for Calderón- Zygmund type operators, Contemporary Mathematics, 446, 2007), [2] S. Chanillo, A note on commutators, Indiana Univ. Math. J., 3, 982), 7-6. [3] W. G. Chen, Besov estimates for a class of multilinear singular integrals, Acta Math. Sinica, 6, 2000), [4] F. Chiarenza M. Frasca, Morrey spaces Hardy-Littlewood maximal function, Rend. Mat., 7, 987), [5] R. Coifman R. Rocherg, Another characterization of BMO, Proc. Amer. Math. Soc., 79, 980), [6] R. R. Coifman, R. Rocherg, G. Weiss, Fractorization theorems for Hardy spaces in several variales, Ann. of Math., 03, 976), [7] G. Di FaZio M. A. Ragusa, Commutators Morrey spaces, Boll. Un. Mat. Ital., 5-A7), 99), [8] G. Di Fazio M. A. Ragusa, Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients, J. Func. Anal., 2, 993), [9] J. Garcia-Cuerva J. L. Ruio de Francia, Weighted norm inequalities related topics, North-Holl Math., Amsterdam, 6, 985. [0] S. Janson, Mean oscillation commutators of singular integral operators, Ark. Math., 6, 978), [] Y. Lin, Sharp maximal function estimates for Calderón-Zygmund type operators commutators, Acta Math. Scientia, 3A), 20), [2] L. Z. Liu, Interior estimates in Morrey spaces for solutions of elliptic equations weighted oundedness for commutators of singular integral operators, Acta Math. Scientia, 25B),), 2005), [3] L. Z. Liu, The continuity of commutators on Trieel-Lizorkin spaces, Integral Equations Operator Theory, 49, 2004), [4] L. Z. Liu, Trieel-Lizorkin space estimates for multilinear operators of sulinear operators, Proc. Indian Acad. Sci. Math. Sci), 3, 2003), [5] L. Z. Liu, Boundedness for multilinear Littlewood-Paley operators on Trieel- Lizorkin spaces, Methods Applications of Analysis, 04), 2004), [6] S. Z. Lu, Four lectures on real H p spaces, World Scientific, River Edge, NI, 995. [7] T. Mizuhara, Boundedness of some classical operators on generalized Morrey spaces, Harmonic Analysis, Proceedings of a conference held in Sendai, 990), [8] B. Muckenhoupt R. L. Wheeden, Weighted norm inequalities for fractional integral, Trans. Amer. Math. Soc., 92, 974), [9] M. Paluszynski, Characterization of the Besov spaces via the commutator operator of Coifman, Rocherg Weiss, 44, 995), -7. Authenticated Download Date 8//3 9:46 AM

19 Vol. L 202) Sharp Maximal Function Estimates Boundedness for Commutator...5 [20] J. Peetre, On convolution operators leaving L p,λ -spaces invariant, Ann. Mat. Pura. Appl., 72, 966), [2] J. Peetre, On the theory of L p,λ -spaces, J. Func. Anal., 4, 969), [22] C. Pérez, Endpoint estimate for commutators of singular integral operators, J. Func. Anal., 28, 995), [23] C. Pérez R. Trujillo-Gonzalez, Sharp weighted estimates for multilinear commutators, J. London Math. Soc., 65, 2002), [24] E. M. Stein, Harmonic analysis: real variale methods, orthogonality oscillatory integrals, Princeton Univ. Press, Princeton NJ, 993. [25] A. Torchinsky, Real variale methods in harmonic analysis, Pure Applied Math. 23, Academic Press, New York, 986. [26] A. Torchinsky S. Wang, A note on the Marcinkiewicz integral, Colloq. Math., 60/6, 990), Guo Sheng Department of Mathematics Changsha University of Science Technology Changsha 40077, P. R. of China lanzheliu@63.com Huang Chuangxia Department of Mathematics Changsha University of Science Technology Changsha 40077, P. R. of China Liu Lanzhe Department of Mathematics Changsha University of Science Technology Changsha 40077, P. R. of China Received: Accepted: Authenticated Download Date 8//3 9:46 AM