L p bounds for the commutators of singular integrals and maximal singular integrals with rough kernels

Transcription

1 L bounds for the commutators of singular integrals and maximal singular integrals with rough kernels To aear in Transactions of the American Mathematical Society) Yaning Chen Deartment of Alied Mathematics, School of Mathematics and Physics University of Science and Technology Beijing Beijing 00083, The Peole s Reublic of China yaningch@6.com and Yong Ding School of Mathematical Sciences Beijing Normal University Laboratory of Mathematics and Comlex Systems, Ministry of Education Beijing 00875, The Peole s Reublic of China dingy@bnu.edu.cn ABSTRACT The commutator of convolution tye Calderon-Zygmund singular integral oerators with rough kernels.v. Ωx) are studied. The authors established the L < < ) x n boundedness of the commutators of singular integrals and maximal singular integrals with the kernel condition which is different from the condition Ω H S n ). MR000) Subject Classification: 4B0, 4B5, 4B99 Keywords: Commutator, singular integral, Maximal singular integral, rough kernel, BMO, Bony araroduct; The research was suorted by NSF of China No , 37057), NCET of China No. NCET--0574), the Fundamental Research Funds for the Central Universities No. 0CXQT09) and SRFDP of China No ). Corresonding author.

2 Introduction The homogeneous singular integral oerator T Ω is defined by Ωx y) T Ω fx) =.v. fy) dy, R x y n n where Ω L S n ) satisfies the following conditions: a) b) Ω is homogeneous function of degree zero on R n \ {0}, i.e. Ωtx) = Ωx) for any t > 0 and x R n \{0}..) Ω has mean zero on S n, the unit shere in R n, i.e. S n Ωx ) dσx ) = 0..) For a function b L loc R n ), let A be a linear oerator on some measurable function sace. Then the commutator between A and b is defined by [b, A]fx) := bx)afx) Abf)x). In 965, Calderón [5] defined a commutator for the Hilbert transform H and a Lishitz function b, which is connected closely the Cauchy integral along Lischitz curves see also [6]). Commutators have layed an imortant role in harmonic analysis and PDE, for examle in the theory of non-divergent ellitic equations with discontinuous coefficients see [4,,, 8]). Moreover, there is also an interesting connection between the nonlinear commutator, considered by Rochberg and Weiss in [9], and Jacobian maing of vector functions. They have been alied in the study of the nonlinear artial differential equations see [3, 5]). In 976, Coifman, Rochberg and Weiss [4] obtained a characterization of L -boundedness of the commutators [b, R j ] generated by the Reisz transforms R j j =,, n, ) and a BMO function b. As an alication of this characterization, a decomosition theorem of the real Hardy sace is given in this aer. Moreover, the authors in [4] roved also that if Ω LiS n ), then the commutator [b, T Ω ] for T Ω and a BMO function b is bounded on L for < <, which is defined by Ωx y) [b, T Ω ]fx) =.v. bx) by))fy)dy. R x y n n In the same aer, Coifman, Rochberg and Weiss [4] outlined a different aroach, which is less direct but shows the close relationshi between the weighted inequalities of the oerator T and the weighted inequalities of the commutator [b, T ]. In 993, Alvarez, Bagby, Kurtz and Pérez [] develoed the idea of [4], and established a generalized boundedness criterion for the commutators of linear oerators. The result of Alvarez, Bagby, Kurtz and Pérez see [, Theorem.3]) can be stated as follows. Theorem A []) Let < <. If a linear oerator T is bounded on L w) for all w A q, < q < ), where A q denote the weight class of Muckenhout, then for b BMO, [b, T ]f L C b BMO f L. Combining Theorem A with the well-known results by Duoandikoetxea [6] on the weighted L boundedness of the rough singular integral T Ω, we know that if Ω L q S n ) for some q >, then

3 [b, T Ω ] is bounded on L for < <. However, it is not clear u to now whether the oerator T Ω with Ω L \ q> Lq S n ) is bounded on L w) for < < and all w A r < r < ), Hence, if Ω L \ q> Lq S n ), the L boundedness of [b, T Ω ] can not be deduced from Theorem A. The urose of this aer is to give a sufficient condition which contains q> Lq S n ), such that the commutator of convolution oerators are bounded on L R n ) for < <, and this condition was introduced by Grafakos and Stefanov in [3], which is defined by where α > 0 is a fixed constant. It is well known that ) +α su Ωy) dσy) <,.3) ξ S n S ξ y n L q S n ) L log + LS n ) H S n ). q> Let F α S n ) denote the sace of all integrable functions Ω on S n satisfying.3). The examles in [3] show that there is the following relationshi between F α S n ) and H S n ) the Hardy sace on S n ) q> L q S n ) α>0 F α S n ) H S n ) α>0 F α S n ). The condition.3) above have been considered by many authors in the context of rough integral oerators. One can consult [, 7, 8, 9, 0, 7, 4] among numerous references, for its develoment and alications. Now let us formulate our main results as follows. Theorem. Let Ω be a function in L S n ) satisfying.) and.). If Ω F α S n ) for some α >, then [b, T Ω ] extends to a bounded oerator from L into itself for α+ α < < α +. Corollary. Let Ω be a function in L S n ) satisfying.) and.). If Ω α> F αs n ), then [b, T Ω ] extends to a bounded oerator from L into itself for < <. The roof of this result is in Section 4. In the roof of Theorem, we have used Littlewood-Paley decomosition and interolation theorem argument to rove L < < ) norm inequalities for rough commutator [b, T Ω ]. These techniques have been used to rove the L < < ) norm inequalities for rough singular integrals in [3] or [5]. They are very similar in sirit, though not in detail. In the following, we will oint out the difference in the methods used to rove L < < ) norm inequalities for rough commutators and rough singular integrals. Let T be a linear oerator, we may decomose T = l Z T l by using the roerties of Littlewood- Paley functions and Fourier transform, reduce T to a sequence of comosition oerators {T l } l Z. Hence, to get the L < < ) norm of T, it suffices to establish the delicate L < < ) norm of each T l with a summation convergence factor, which can be obtained by interolating between the delicate L norm of T l, which has a summation convergent factor, and the L q < q < ) norm of T l, for each l Z. Let T be a rough singular integral. The delicate L norm of each T l can be obtained by using Fourier transform, the Plancherel theorem and the Littlewood-Paley theory. The L q < q < ) norm of each 3

4 T l can be obtained by the method of rotations, the L q < q < ) bounds of the one dimensional case of Hardy-Littlewood oerator and the Littlewood-Paley theory. On the other hand, if T is a rough commutator of singular integral, the delicate L norm of each T l can be obtained by using the L norm of the commutators of Littlewood-Paley oeratorssee Lemma 3.3) and Lemma 3.4 in Section 3. With these techniques and lemmas, G. Hu [6] obtained the result in Theorem for =. Therefore, it reduces the L < < ) norm of T to the L q < q < ) norm of T l for each l Z. Unfortunately, since each T l is generated by a BMO function and a comosition oerator, the method of rotations, which deals with the same roblem in rough singular integrals, fails to treat this roblem directly. Hence we need to look for a new idea. We find the Bony araroduct is the key technique to resolve the roblem. In articular, it is worth to oint out that main method used in this aer gives indeed a new alication of Bony araroduct. It is well known that the Bony araroduct is an imortant tool in PDE. However, the idea resented in this aer shows that the Bony araroduct is a owerful tool also for handling the integral oerators with rough kernels in harmonic analysis. It is well known that maximal singular integral oerators TΩ lay a key role in studying the almost everywhere convergence of the singular integral oerators. The maing roerties of the maximal singular integrals with convolution kernels have been extensively studied see [5, 3, 30], for examle). Therefore, another aim of this aer is to give the L R n ) boundedness of the maximal commutator [b, TΩ ] associated to the singular integral T Ω, which is defined by [b, TΩ]fx) = su Ωx y) bx) by))fy) dy x y > x y n. j The following theorem is another main result given in this aer: Theorem. Let Ω be a function in L S n ) satisfying.) and.). If Ω F α S n ) for some α >, then [b, T Ω ] extends to a bounded oerator from L into itself for α α < < α. Corollary. Let Ω be a function in L S n ) satisfying.) and.). If Ω α> F αs n ), then [b, T Ω ] extends to a bounded oerator from L into itself for < <. One will see that the maximal commutator [b, TΩ ] can be controlled ointwise by some comosition oerators of T Ω, M, M Ω and their commutators [b, T Ω ], [b, M] and [b, M Ω ], where M is the standard Hardy-Littlewood maximal oerator, M Ω defined by denotes the maximal oerator with rough kernel, which is M Ω fx) = su Ωx y) j < x y x y n fy)dy. j+ The corresonding commutators [b, M] and [b, M Ω ] are defined by [b, M]fx) = su r>0 r n bx) by) fy) dy and x y <r [b, M Ω ]fx) = su Ωx y) bx) by)) j < x y < x y n fy)dy. j+ We give the following L R n ) boundedness of the commutators [b, M Ω ]: 4

5 Theorem 3. Let Ω be a function in L S n ) satisfying.). If Ω F α S n ) for some α >, then [b, M Ω ] extends to a bounded oerator from L into itself for α+ α < < α +. Corollary 3. Let Ω be a function in L S n ) satisfying.). If Ω α> F αs n ), then [b, M Ω ] extends to a bounded oerator from L into itself for < <. Theorem 3 is actually a direct consequence of the L R n ) boundedness of the commutator formed by a class of Littlewood-Paley square oerator with rough kernel and a BMO function. Ω = Ω A S n with A = S n Ωx )dσx ), then Ω satisfies.). It is easy to check that [b, M Ω ]fx) su bx) by)) Ωx y) fy) dy j < x y < x y n + C[b, M]fx) j+ C[b, g Ω]fx) + [b, M]fx)), In fact, if where g Ω and [b, g Ω ] denote the Littlewood-Paley square oerator and its commutator, which are defined resectively by and g Ω fx) = Ωx y) j < x y x y n fy)dy j+ ) / [b, g Ω ]fx) = Ωx y) bx) by)) j < x y < x y n fy)dy / ). j+ Thus,.4) shows that Theorem 3 will follow from the L R n ) boundedness of the commutators [b, g Ω ] and [b, M]. Since the L R n ) boundedness of the later is well known see []), hence, we need only give the L R n ) boundedness of the commutator [b, g Ω ] which can be stated as follows. Theorem 4..4) Let Ω be a function in L S n ) satisfying.) and.). If Ω F α S n ) for some α >, then [b, g Ω ] extends to a bounded oerator from L into itself for α+ α < < α +. Corollary 4. Let Ω be a function in L S n ) satisfying.) and.). If Ω α> F αs n ), then [b, g Ω ] extends to a bounded oerator from L into itself for < <. In fact, Theorem 4 is a corollary of Theorem. Write T Ω fx) = K j fx), where K j x) = Ωx) x n χ {j < x j+ }. Define T j fx) = K j fx), then [b, T Ω ]fx) = [b, T j]fx), and [b, g Ω ]fx) = [b, T j]fx) ) /. Then we get the L boundedness of [b, g Ω ] by using Theorem, Rademacher function and Khintchine s inequalities. This aer is organized as follows. First, in Section, we give some imortant notations and tools, which will be used in the roofs of the main results. In Section 3, we give some lemmas which will be used in the roofs of the main results. In Section 4, we rove Theorem by alying the lemmas in Section 3. Finally, we rove Theorem by alying Theorem 3 and Theorem 4 in Section 5. Throughout this aer, the letter C will stand for a ositive constant which is indeendent of the essential variables and not necessarily the same one in each occurrence. 5

6 Notations and reliminaries Let us begin by giving some notations and imortant tools, which will be used in the roofs of our main results.. Schwartz class and Fourier transform. Denote by S R n ) and S R n ) the Schwartz class and the sace of temered distributions, resectively. The notations and denote the Fourier transform and the inverse Fourier transform, resectively.. Smooth decomosition of identity and multiliers. Let ϕ S R n ) be a radial function satisfying 0 ϕ with its suort is in the unit ball and ϕξ) = for ξ. The function ψξ) = ϕ ξ ) ϕξ) S R n ) suorted by { ξ } and satisfies the identity ψ j ξ) =, for ξ 0. For j Z, denote by j and G j the convolution oerators whose the symbols are ψ j ξ) and ϕ j ξ), resectively. That is, j and G j are defined by j fξ) = ψ j ξ) ˆfξ) and Ĝjfξ) = ϕ j ξ) ˆfξ). By the Littlewood-Paley theory, for < < and {f j } L l ), the following vector-value inequality holds see [,.343]) j+k f j ) / L C 3. Homogeneous Triebel-Lizorkin sace F s,q and s R, the homogeneous Triebel-Lizorkin sace f j ) / L, for k [ 0, 0]..) R n ) and Besov sace Ḃs,q R n ). For 0 <, q ) F s,q R n ) is defined by { F s,q R n ) = f S R n ) : f F s,q = jsq j f q)/ql and the homogeneous Besov sace Ḃs,q R n ) is defined by } < { ) /q } Ḃ s,q R n ) = f S R n ) : f Ḃs,q = jsq j f q L <, where S R n ) denotes the temered distribution class on R n. 4. Sequence Carleson measures. A sequence of ositive Borel measures {v j } is called a sequence Carleson measures in R n Z if there exists a ositive constant C > 0 such that j k v jb) C B for all k Z and all Euclidean balls B with radius k, where B is the Lebesgue measure of B. The norm of the sequence Carleson measures v = {v j } is given by { } v = su v j B), B where the suremum is taken over all k Z and all balls B with radius k. 5. Homogeneous BMO-Triebel-Lizorkin sace. For s R and q < +, the homogeneous BMO - Triebel-Lizorkin sace F s,q is the sace of all distributions b for which the sequence { sjq j b)x) q dx} is a Carleson measure see [9]). The norm of b in F s,q is given by [ b F s,q = su B j k B j k ] sjq j b)x) q q dx 6

7 where the suremum is taken over all k Z and all balls B with radius k. F s, = Ḃs,. Moreover, F 0, = BMO see [9, 0 ]. For q = +, we set 6. Bony araroduct and Bony decomosition. The araroduct of Bony [3] between two functions f, g is defined by π f g) = j f)g j 3 g). At least formally, we have the following Bony decomosition 3 Lemmas fg = π f g) + π g f) + Rf, g) with Rf, g) = i Z k i i f) k g)..) We first give some lemmas, which will be used in the roof of Theorem and Theorem. Riesz otential and its inverse. For 0 < τ < n, the Riesz otential I τ of order τ is defined on S R n ) by setting Îτ fξ) = ξ τ fξ). Another exression of I τ is fy) I τ fx) = γτ) dy, R x y n τ n where γτ) = τ π n/ Γ n τ )/Γ τ ). Moreover, for 0 < τ < n, the inverse oerator I τ of I τ is defined by Î τ fξ) = ξ τ fξ), where denotes the Fourier transform. With the notations above, we show the following two facts: Lemma 3. For 0 < τ < /, we have γτ) Cτ, 3.) where C is indeendent of τ. Proof. Alying the Stirling s formula, we have πx x / e x Γx) πx x / e x for x >. Thus, by the equation sγs) = Γs + ) for s > 0, we get Γ n τ ) = n τ n τ Γ + ) ) n τ n τ π + + ) e n τ) n τ C 3.) and Γ τ ) = τ Γ τ + ) ) τ τ π + + ) e τ τ C/τ. 3.3) Hence, 3.) follows from 3.) and 3.3). Obviously, the constant C in 3.) is indeendent of τ. Lemma 3. For the multilier G k k Z), b BMOR n ), and any fixed 0 < τ < /, we have where C is indeendent of k and τ. G k bx) G k by) C kτ τ x y τ b BMO, 3.4) 7

8 Proof. Note that I τ Iτ f) = f, we have G k bx) = γτ) Rn Iτ G k b)z) x z n τ dz. Hence We first show that In fact, G k bx) G k by) = γτ) z 3 x y Iτ R n G k b)z) γτ) Iτ G k b) L = γτ) Iτ G k b) L R x y + z n τ z n τ dz n = z x y x y + z n τ dz + z n τ C x y τ, τ R n R n ) x z n τ y z n τ dz x z n τ y z n τ dz x y + z n τ z n τ dz. 3.5) x y + n τ n τ Cτ x y τ. 3.6) L z n τ z x y dz + z > x y dz + C z n τ where C is indeendent of τ. By 3.5), 3.6) and 3.), we get where C is indeendent of τ. We now estimate Iτ Iτ G k b) L = I τ G k u b) L u Z x y + z n τ z > x y x y dz z n τ+ z n τ dz G k bx) G k by) C x y τ I τ G k b) L, 3.7) u k+ G k b) L. Since G k u b = 0 for u k +, we have Gk Iτ u b) L u k+ I τ u b L. 3.8) Taking a radial function ψ S R n ) such that su ψ) {/4 x 4} and ψ = in {/ x }. Then we have u bξ) = uτ ψ u ξ) u ξ τ u bξ). Iτ Set a function h by ĥξ) = ψξ) ξ τ. Then I τ u bx) = uτ R n un h u x y)) u by) dy. So we have I τ u b L uτ un h u ) L u b L = uτ h L u b L. Thus, if there exists a constant C > 0, indeendent of τ, such that h L C, 3.9) 8

9 then by 3.7)-3.8), we have Since for some 0 < τ <, u k+ G k bx) G k by) C x y τ u k)τ = j= u k+ uτ u b L C x y τ kτ su u b L u Z u k+ u k)τ. jτ τ τ = = τ τ = τ τ θτ < C, for 0 < τ < /, τ where C is indeendent of τ. Using the fact see [,.65]) we have su u b L C n b BMO, 3.0) u Z G k bx) G k by) C x y τ kτ τ b BMO, where C is indeendent of k and τ. Thus, to finish the roof of Lemma 3., it remains to show 3.9). In fact, h L = x < hx) dx + hx) dx C n h L + n h ) L ) := Cn I + I ). x Since su ψ) {/4 ξ 4} and 0 < δ < /, we get I = ψξ) ξ τ L C, where C is indeendent of τ. Thus, to get 3.9), we need only verify that I us recall some notations about the multi-index. C. To do this, let For a multi-index α = α,..., α n ) Z n +, denote α f = α... αn n f, α = α + + α n and x α = x α... xαn n for x R n. By [,.45], we know that and the function m α ξ) = + ξ ) n/ = ξ α + ξ ) n/ + ξ ) n/ hξ) ) = α n α n n! α!... α n! ξα + ξ ) n/ is an L < < ) multilier whenever α n. Hence ξ α C α,n m α ξ)ξ α hξ)) = C α n C α,n m α ξ) αȟξ)), where denote the inverse Fourier transform. Alying the equation above, we get I C + ξ ) n/ hξ) L = + ξ ) n/ hξ)) L C C α,n αȟ L α n = C C α,n α ȟ L = C C α,n α ψξ) ξ τ ) L. α n α n Notice that α ψξ) ξ τ ) = β α C β α... C βn α n β ψξ)) α β ξ τ )), 3.) where the sum in 3.) is taken over all multi-indices β with 0 β j α j for all j n. Trivial comutations show that there exists C > 0, indeendent of τ, such that α β ξ τ ) C for /4 < ξ < 4 9

10 and 0 < τ < /. Further, by ψ C0 R n ), then β ψξ) C. So we get α ψξ) ξ τ ) C. From this we get I / C α,n α ψξ) ξ τ ) L dξ) C, α n /4 ξ 4 where C is deendent only on n, but indeendent of τ. This comletes the estimate of 3.9) and Lemma 3. follows. Lemma 3.3 see [7]). Let φ S R n ) be a radial function such that su φ {/ ξ } and l Z φ3 l ξ) = for ξ 0. Define the multilier oerator S l by Ŝlfξ) = φ l ξ) fξ), Sl by Sl f = S ls l f). For b BMOR n ), denote by [b, S l ]resectively, [b, Sl ] ) the commutator of S l resectively, Sl ). Then for < < and f L R n ), we have ) / L i) [b, S l ]f) Cn, ) b BMO f L ; l Z ) / L ii) [b, Sl ]f) Cn, ) b BMO f L ; l Z iii) S l ]f l ) l Z[b, ) / L Cn, ) b BMO f l {f l } L L, l ). l Z Lemma 3.4 see [6]). Let m σ C0 R n )0 < σ < ) be a family of multiliers such that sum σ ) { ξ σ}, and for some constants C, 0 < A /, and α > 0, m σ L C min{aσ, log α + σ)}, m σ L C. Let T σ be the multilier oerator defined by T σ fξ) = m σ ξ) fξ). For b BMO, denote by [b, T σ ] the commutator of T σ. Then for any fixed 0 < v <, there exists a ositive constant C = Cn, v) such that [b, T σ ]f L CAσ) v log/a) b BMO f L, ifσ < 0/ A; [b, T σ ]f L C log α+)v+ + σ) b BMO f L, ifσ 0/ A; Similar to the roof of Lemma 3.4, it is easy to get Lemma 3.5 Let m σ C0 R n )0 < σ < ) be a family of multiliers such that sum σ ) { ξ σ}, and for some constants C, 0 < A /, and α > 0, j N m σ L C min{a j σ, log α + j σ)}, m σ L C j. Let T σ be the multilier oerator defined by T σ fξ) = m σ ξ) fξ). For b BMO, denote by [b, T σ ] the commutator of T σ. Then for any fixed 0 < v <, there exists a ositive constant C = Cn, v),0 < β < such that [b, T σ ]f L C βj Aσ) v log/a) b BMO f L, ifσ < 0/ A; 0

11 Lemma 3.6. [b, T σ ]f L C log α+)v+ + j σ) b BMO f L, ifσ 0/ A; For any j Z, let K j x) = Ωx) x n χ { j < x j+ }x). Suose Ω L S n ) satisfying.). Then for < <, the following vector valued inequality ) / L ) / L K j f j C Ω L f j holds for any {f j } in L l ). Proof. Note that for Ω L S n ) and any local integrable function f on R n, we have where σ f)x) := su K j fx) CM Ω fx) for any x R n, M Ω fx) = su r>0 r n Ωx y) fy) dy. 3.) x y <r By the L q boundedness of M Ω for all q > with Ω L S n ), σ is also a bounded oerator on L q R n ) for all q > with Ω L S n ). Thus, by alying Lemma in [5,.544], we know that, for < <, the vector valued inequality 3.) holds. Lemma 3.7. For any j Z, define the oerator T j by T j f = K j f, where K j x) = Ωx) x n χ {j < x j+ }x). Denote by [b, S l j T j S l j ] the commutator of S l jt j S l j. Suose Ω L S n ) satisfying.). Then for any fixed 0 < τ < /, b BMOR n ), < <, [b, S l j T j Sl j]f where C is indeendent of τ and l. Proof. For any j, l Z, we may write L C b BMO max{ τl τ, } Ω L f L, 3.3) [b, S l j T j S l j ]f = [b, S l j]t j S l j f) + S l j[b, T j ]S l j f) + S l jt j [b, S l j ]f). Thus, [b, S l j T j Sl j]f L [b, S l j ]T j Sl jf) + S l j T j [b, S L l j]f) + S l j [b, T j ]Sl jf) := L + L + L 3. L L 3.4) Below we shall estimate L i for i =,, 3, resectively. For L, by Lemma 3.3 iii), Lemma 3.6 and the Littlewood-Paley theory, we have L C b BMO T j Sl jf ) / L C Ω L b BMO Sj f ) / L C Ω L b BMO f L. Similarly, we have L C Ω L b BMO f L.

12 Hence, by 3.4), to show 3.) it remains to give the estimate of L 3. We will aly the Bony araroduct to do this. By.), we have Thus [b, T j ]S l j fx) = bx)t js l j f)x) T jbs l j f)x) = [π TjS l j f) b)x) T j π S l j f)b))x)] + [Rb, T j S l j f)x) T jrb, S l j f))x)] +[π b T j S l j f)x) T jπ b S l j f))x)]. L 3 [ S l j πtjs l j f) b) T j π S l j f)b)) ] L [ + S l j Rb, Tj Sl jf) T j Rb, Sl jf)) ] L [ + S l j πb T j Sl jf) T j π b Sl jf)) ] L := M + M + M 3. a) The estimate of M. For M, by i S l j g = 0 for g S R n ) when i l j) 3, we get Note that π TjSl j f) b)x) T j π S l j f)b))x) = {T j i Sl jf)x)g i 3 b)x) T j [ i Sl jf)g i 3 b)]x)} = i l j) i l j) [G i 3 b, T j ] i S l j f)x) By Lemma 3., we have [G i 3 b, T j ] i S l j fx) [G i 3 b, T j ] i S l jf)x). = Ωx y) j x y < x y n G i 3bx) G i 3 by)) i Sl jfy)dy j+ Ωx y) C j x y < x y n G i 3bx) G i 3 by) i Sl jfy) dy. j+ C iτ τ b Ωx y) BMO j x y < x y n x y τ i Sl jfy) dy j+ C i+j)τ Ωx y) b BMO τ j x y < x y n isl jfy) dy j+ 3.5) 3.6) 3.7) 3.8) = C i+j)τ b BMO T Ω,j i S τ l jf )x), where Ωx y) T Ω,j fx) = j x y < x y n fy)dy. j+ Then, by 3.6), 3.8) and alying Lemma 3.6,.) and the Littlewood-Paley theory, we have that, for any fixed 0 < τ < /, M C τl τ b BMO ) /L T Ω,j l j+k Sl jf ) k C τl τ b BMO Ω L j+k Sj f ) / L k C τl τ b BMO Ω L S j f ) / L C τl τ b BMO Ω L f L, 3.9)

13 where C is indeendent of l and τ. b) The estimate of M. Since k, i+k S l j g = 0 for g S R n ) when i l j) 8. Thus Rb, T j S l j f) T j Rb, S l j f))x) = i b)x)t j i+k S l j f)x) T j i Z = = k k= i l j) 7 k= i l j) 7 i Z k ) i b) i+k S l j f) x) i b)x)t j i+k S l j f)x) T j i b) i+k S l j f) ) x) [ i b, T j ] i+k S l j f)x). By the equality above and using Lemma 3.6,.), 3.0) and the Littlewood-Paley theory, we have M C Ω L su i b) L T j, Ω l j+k S l j f ) ) / L i Z k 7 C b BMO Ω L Sj f ) / L C b BMO Ω L f L. ) 3.0) c) The estimate of M 3. Finally, we give the estimate of M 3. Note that S l j i g)g i 3 h) ) = 0 for g, h S R n ) if i l j) 5. Thus we get S l j πb T j S l j f) T j π b S l j f)) ) = S l j i b)g i 3 T j S l j f) T j i b)g i 3 S l j f) )) x) i Z i Z { = S l j i b)g i 3 T j S l j f) ) x) S l j T j i b)g i 3 S l j f) ) } x) = i l j) 4 i l j) 4 S l j [ i b, T j ]G i 3 S l j f) ). Alying Proosition 5..4 in [,.343], it is easy to see that G j+k S j f j ) / L f j ) / L for k [ 0, 0]. Thus, by the Littlewood-Paley theory, Lemma 3.6 and 3.0) we get M 3 C su i b) L T Ω,j G l j+k 3 S l j f ) ) / L i Z k 4 C b BMO Ω L S j f ) / L C b BMO Ω L f L. 3.) By 3.5), 3.9)-3.), we get L 3 C max{, τl τ } b BMO Ω L f L for l Z. Combining this with 3.4), we comlete the roof of 3.3). 3

14 4 Proof of Theorem Let φ C0 R n ) be a radial function such that 0 φ, suφ {/ ξ } and φ 3 l ξ) =, ξ 0. l Z Define the multilier oerator S l by Ŝ l fξ) = φ l ξ) fξ). Let K j x) = Ωx) x n χ { j < x j+ }. Define the oerator and the multilier T l j by T l j fξ) = Ωy) T j fx) = K j fx) = fx y) dy, j < y y n j+ T j S l j fξ) = φ j l ξ) K j ξ) fξ). With the notations above, it is easy to see that [b, T Ω ]fx) = S l j T j Sl j]fx) l Z [b, = S l j TjS l Z [b, l l j ]fx) := V l fx), l Z where V l fx) = [b, S l j T l js l j ]fx). Then by the Minkowski inequality, we get [b, T Ω ]f L [log ] l= V l f + L l=[log ]+ V l f 4.) L. Now, we will estimate the two cases resectively. [log ] Case. The estimate of V l f L. l= Since Ω L S n ) satisfies.) and.), by a well-known Fourier transform estimate of Duoandikoetxea and Rubio de Francia See [5,.55-55]), it is easy to show that K j ξ) C Ω L j ξ. A trivial comutation gives that K j L C j Ω L. Set m j ξ) = K j ξ), m l j ξ) = m jξ)φ j l ξ), and recall that T l j by T l j fξ) = ml jξ) fξ). Straightforward comutations lead to m l j j ) L C Ω L l, m l j j ) L C Ω L, su{m l j j ξ)} { ξ l+ }. Let T l j be the oerator defined by T l j fξ) = ml j j ξ) fξ). 4

15 Denote by T l j;b, f = [b, T l j ]f and T l j;b,0 f = T l j f. Similarly, denote by T l j;b, f = [b, T l j ]f and T l j;b,0 f = T l j f. Thus via the Plancherel theorem and Lemma 3.4 states that for any fixed 0 < v <, k {0, }, T l j;b,kf L C b k BMO Ω L vl f L, l [log ]. Dilation-invariance says that T l j;b,kf L C b k BMO Ω L vl f L, l [log ]. 4.) First, we will give the L -norm estimate of V l f by using the inequality 4.). Recalling that V l fx) = [b, S l j TjS l l j ]fx), for any j, l Z, we may write [b, S l j T l j S l j]f = [b, S l j ]T l j S l jf) + S l j [b, T l j ]S l jf) + S l j T l j [b, S l j]f). Thus, V l f L S l j ]TjS [b, l l j f) + S l j T l L j[b, S l j ]f) + S l j [b, Tj]S l l j f) := Q + Q + Q 3. For Q, by Lemma 3.3iii), 4.) for k = 0 and the Littlewood-Paley theory, we get L L 4.3) Q C b BMO T l js l j f ) /L C b BMO vl ) /L Ω L S l j f C b BMO vl Ω L f L. 4.4) For Q, by the Littlewood-Paley theory, 4.) for k = 0 and Lemma 3.3i), we get Q ) /L C Tj[b, l S l j ]f) C vl ) /L Ω L [b, S l j ]f C b BMO vl Ω L f L. 4.5) About Q 3, by4.) for k = and the Littlewood-Paley theory, we have Q 3 ) /L C [b, Tj]S l l j f) C vl ) /L Ω L S l j f C b BMO vl Ω L f L. 4.6) Combining 4.4) with 4.5) and 4.6), we have V l f L C b BMO vl Ω L f L, l [log ]. 4.7) 5

16 On the other hand, since T l j fx) = T js l j fx), then V l fx) = [b, S l j T j S l j]fx). Alying Lemma 3.7, we get for < < V l f L C b BMO Ω L f L, l [log ]. 4.8) Interolating between 4.7) and 4.8), there exists a constant 0 < β <, such that V l f L C βvl Ω L b BMO f L, l [log ]. 4.9) Then by the Minkowski inequality, we get for < < [log ] l= Case. The estimate of V l f L l=+[log ] C [log ] l= [log ] l= V l f L βvl b BMO Ω L f L C b BMO Ω L f L. V l f L. 4.0) Recalling that V l fx) = [b, S l j T j S l j]fx). We will give the delicate L norm of V l f and the L < < ) norm of V l f resectively. It is easy to see that if Ω F α S n ) for α > satisfies.) and.), K j ξ) C log α j ξ + ), K j L C j. Set m j ξ) = K j ξ), m l j ξ) = φj l ξ)m j ξ). Let T l j Straightforward comutations lead to be the oerator defined by T j lfξ) = ml j ξ) fξ). m l j j ) L C log α + l ), m l j j ) L C, su{m l j j ξ)} { ξ l+ }. Let T l j be the oerator defined by T l j fξ) = ml j j ξ) fξ). Denote by Tj;b, l f = [b, T j l]f and T j;b,0 l f = T j lf. Similarly, denote by T j;b, l f = [b, T j l]f and T j;b,0 l f = T j lf. Thus via the Plancherel theorem and Lemma 3.4 with σ = l states that for any fixed 0 < v <, k {0, }, T j;b,kf l L C b k BMOC log α )v+ + l ) f L, l + [log ]. 4.) Dilation-invariance says that Tj;b,kf l L C b k BMO log α )v+ + l ) f L, l + [log ]. 4.) Alying 4.), Lemma 3.3 and the Littlewood-Paley theory, similar to the roof of 4.7), we get V l f L C b BMO log α )v+ + l ) f L, l + [log ], 4.3) 6

17 On the other hand, by Lemma 3.7, for any fixed 0 < τ < /, < <, τl V l f L C b BMO τ Ω L f L, l + [log ], where C is indeendent of τ and l. Take τ = /l, we get V l f L Cl b BMO Ω L f L, l + [log ], where C is indeendent of l. Which says that for any r satisfying < r <, we have V l f L r Cl b BMO f L r, l + [log ]. 4.4) Now for any, we take r sufficient large such that r >. Using the Riesz-Thorin interolation theorem between 4.3) and 4.4), we have that for any l + [log ], V l f L C b BMO l θ log α )v+)θ + l ) f L, where θ = r ) r ). We can see that if r, then θ goes to / and log α )v+)θ + l ) goes to log α )v+)/ + l ). Therefore, we get V l f L C b BMO l / log α )v+) + l ) f L, l + [log ],. 4.5) Then by the Minkowski inequality, for < α +, we get V l f l=+[log ] C b BMO L l=+[log ] C b BMO f L. If < <, by duality, we get for > α+ α l=+[log ] V l f L l / l α )v+) f L Combining 4.6) with 4.7), we get for α+ α < < α +, V l f C b BMO f L. l=+[log ] This comletes the roof of Theorem. L 4.6) C b BMO f L. 4.7) 5 Proof of Theorem Let α >, K j and the oerator T j be the same as in the roof of Theorem. Define [b, TΩ s]fx) = Ωx y) bx) by)) fy) dy x y > x y n s Ωx y) = bx) by)) fy) dy j < x y x y n j+ = [b, T j ]fx), 7

18 where So, we get Ωx y) [b, T j ]fx) = bx) by)) fy) dy. j x y x y n 5.) j+ su [b, TΩ s]fx) s>0 su [b, T j ]fx), Thus, to rove Theorem, it suffices to estimate the L norm of su [b, T j ]fx). Taking a radial Schwartz function Φ such that Φξ) = for ξ and Φξ) = 0 for ξ >, and define Φ s Φ s ξ) = Φ s ξ). Write [b, T j ]fx) = Observed that [ Φ s [b, T Ω ]f := L s fx) + J s fx). s Φ s j= s j= ) [b, T j ]f x) = [b, Φ s ) ] [ [b, T j ]f x) + [b, T j ]fx) Φ s s j= s K j ]fx) [b, W s ] j= where W s is a convolution oerator with its convolution kernel Φ s. Observe that see [5]) and s j= Φ s s j= T j fx) = T Ω fx) K j x) C Ω L ns / + s x n+ ) T j fx). It follows that [b, T j ]f ) T j f x), su L s fx) CM[b, T Ω ]f)x) + C[b, M]fx) + [b, M]T Ω f)x) + [b, M]TΩf)x). ) ] x) Then by Theorem, the L α α < < α) boundedness of T Ω, T Ω with kernel function Ω F α for α > see [3]) and [b, M] see []), we get for To estimate su J s fx), write ) [b, T j ]fx) Φ s [b, T j ]f x) = Thus we get su α α < < α, su L s f L C b BMO f L. 5.) [b, T j ]fx) [b, Φ s K j ]fx) + [b, W s ] J s fx) su [b, δ Φ s ) K j ]fx) + [b, M]TΩf)x), where δ is Dirac mass at the origin. Since for α α < < α, see [3]) ) T j f x). [b, M]T Ωf) L C b BMO f L. 5.3) by 8

19 Thus, to give the estimate of L norm for the term su J s fx), it suffices to give the estimate of L norm for the term su [b, δ Φ s ) K j ]fx). Note that su [b, δ Φ s ) K j ]fx) j=0 su [b, δ Φ s ) K j+s ]fx). Let U s,j fx) = δ Φ s ) K s+j f and [b, U s,j ]fx) = [b, δ Φ s ) K s+j ]f. Then It is easy to see that su [b, δ Φ s ) K j ]fx) su [b, U s,j ]fx) C su [b, T s+j ]fx) + C su j=0 su [b, U s,j ]fx). 5.4) ) W s [b, T s+j ]f + C[b, W s ]T s+j f) x) C su [b, T s+j ]fx) + CMsu C[b, M Ω ]fx) + CM[b, M Ω ]f)x) + C[b, M]M Ω f)x). [b, T s+j ]f )x) + C[b, M]M Ω f)x) Alying Theorem 3, the L < < ) boundedness of M, M Ω with kernel function Ω L S n ) see []) and [b, M] see []), we have for On the other hand, set α α < < α, su [b, U s,j ]f L C [b, M Ω ]f L + b BMO M Ω f L ) C b BMO f L. 5.5) B s,j ξ) = Φ s ξ)) K s+j ξ), B l s,jξ) = Φ s ξ)) K s+j ξ)φ s l ξ). Define the oerator U l s,j by Û l s,j fξ) = Û s,j fξ)φ s l ξ), and denote by [b, U l s,j ] the commutator of U l s,j. Then it is clear that [b, U s,j ]fx) = l Z[b, U l s,js l s]fx). By the Minkowski inequality, we get su [b, U s,j ]f L ) / L [b, U s,j ]f ) / L [b, Us,jS l l s]f l Z ) / L [b, Us,j]S l l sf l Z + ) / L Us,j[b, l Sl s]f l Z := I + I. 5.6) To comlete the roof we will estimate each term searately. Denote by Us,j;b, l f = [b, U s,j l ]f and Us,j;b,0 l f = U s,j l f. Obviously, if we can rove that for any 0 < v <, k {0, }, there exists a constant 0 < β <, such that U l s,j;b,kf L C βj b k BMO l f L, for l [log ] 5.7) 9

20 and U l s,j;b,kf L C b k BMO log α )v+ l+j + ) f L, for l [log ], ) then we may finish the estimate of I and I. We consider first I. In fact, by 5.7) and 5.7 ) for k = and the Littlewood-Paley theory, we get I [log ] l= ) / L [b, Us,j]S l l sf + C βj b BMO [log +C b BMO ] l= l=[log ]+ l=[log ]+ ) / L [b, Us,j]S l l sf ) / L l Sl sf ) / L log α )v+ l+j + ) Sl sf ) ). Since l + j) lj + ), we get I Cj + ) α )v+ b BMO f L. 5.8) We will now estimate I. by 5.7) for k = 0, the Littlewood-Paley theory and Lemma 3.3 ii), we get I [log ] l= C βj [log ) / L Us,j[b, l Sl s]f + ] l= +C l=[log ]+ l=[log ]+ ) / L Us,j[b, l Sl s]f ) / L l [b, Sl s]f ) / ) L log α )v+ l+j + ) [b, Sl s]f Cj + ) α )v+ b BMO f L. ) 5.9) Combining I with I, we get su [b, U s,j ]f L Cj + ) α )v+ b BMO f L. 5.0) Interolating between 5.5) and 5.0), similar to the roof of 4.5), for, we get su [b, U s,j ]f L Cj + ) α )v+ b BMO f L. 5.) Then by 5.4), we get for < α, su [b, δ Φ s ) K j ]fx) Similarly, for <, we get L j + ) j=0 C b BMO f L. α )v+ b BMO f L 5.) su [b, U s,j ]f L Cj + ) α )v+ b BMO f L. 5.3) 0

21 α Then by 5.4), we get for α < <, su [b, δ Φ s ) K j ]fx) L j + ) α )v+ b BMO f L j=0 C b BMO f L. 5.4) This comletes the roof of Theorem. Hence it remains to rove 5.7) and 5.7 ). To this end, define multilier Ũ l s,j by Ũ l s,j fξ) = Bl s,j s ξ) fξ), and denote by [b, Ũ l s,j ] the commutator of Ũ l s,j. Define Ũs,j;b, l f = [b, Ũ s,j l ]f and Ũ s,j;b,0 l f = Ũ s,j l f. Recall that B s,j ξ) = Φ s ξ)) K s+j ξ), B l s,jξ) = Φ s ξ)) K s+j ξ)φ s l ξ). It ie easy to see that B s,j ξ) C j s ξ for s ξ, B s,j ξ) C log α s+j ξ + ) for s ξ >, B s,j ξ) C s j. Since sub l s,j s ξ)) {ξ : l ξ l }, we have the following estimates B s,j s ξ) C l j for l 0 B s,j s ξ) C log α l+j + ) for l > 0, Bs,j l s ξ) C j, Alying Lemma 3.5 with σ = l, A = / and the Plancherel theory, there exists a constant 0 < β <, such that for any fixed 0 < v <, k {0, }, Ũ s,j;b,kf l L C b k BMO βj l f L, for l [log ]. Which imlies that Ũ l s,j;b,kf L C b k BMO log α )v+ l+j + ) f L, for l [log ] +. U l s,j;b,kf L C b k BMO βj l f L, for l [log ]. U l s,j;b,kf L C b k BMO log α )v+ l+j + ) f L, for l [log ] +, by dilation invariance. This establishes the roof of 5.7) and 5.7 ). References [] H. Al-Qassem and A. Al-Salman, Rough Marcinkiewicz integral oerators, Int. J. Math. Math. Sci., 7 00),

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