Lin and Zhang Journal of Inequalities and Applications 07 07:76 DOI 0.86/s3660-07-553- R E S E A R C H Open Access Sharp maximal and weighted estimates for multilinear iterated commutators of multilinear integrals with generalized kernels Yan Lin * and Nan Zhang * Correspondence: linyan@cumtb.edu.cn School of Sciences, China University of Mining and echnology, eijing, 00083, P.R. China Abstract In this paper, the authors establish the sharp maximal estimates for the multilinear iterated commutators generated by MO functions and multilinear singular integral operators with generalized kernels. As applications, the boundedness of this kind of multilinear iterated commutators on the product of weighted Lebesgue spaces and the product of variable exponent Lebesgue spaces can be obtained, respectively. MSC: 40; 45; 435 Keywords: multilinear singular integral operator; multilinear iterated commutator; MO function; sharp maximal function Introduction he multilinear singular integral operator theory plays an important role in the singular integral operator theory of harmonic analysis. On the one hand, many researchers have put plenty time and energy into this topic. Kenig and Stein did many works on the multilinear fractional integral operator in []. Grafakos and orres established the multilinear Calderón-Zygmund theory in []. he boundedness of multilinear Calderón-Zygmund operators with kernel of Dini s type was studied by Lu and Zhang in [3]. On the other hand, more and more researchers have been interested in multilinear commutators. he multilinear commutator was given by Pérez and rujillo-gonzález in [4]. he weighted estimate for multilinear iterated commutators of multilinear fractional integrals was studied by Si and Lu in [5]. Some useful conclusions of multilinear singular integral operators with generalized kernelsweregivenbylinandxiaoin[6]. With more weaker conditions for the kernel, they got the conclusion on sharp maximal estimates of the multilinear singular integral operators and their multilinear commutators with MO functions. Moreover, the boundedness of themultilinear commutatorswith MO functions on the product of weighted Lebesgue spaces and the product of variable exponent Lebesgue spaces was acquired in [6] as well. Pérez, Pradolini, orres and rujillo-gonzález studied the multilinear iterated commutators of multilinear singular integrals with Calderón-Zygmund kernels in [7]. hey first established the sharp maximal estimates, then the end-point estimates were acquired. ased on these studies above, we will focus on the multilinear iterated commutators of multilinear singular integrals with generalized kernels in this paper. And we will con- he Authors 07. his article is distributed under the terms of the Creative Commons Attribution 4.0 International License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Linand ZhangJournal of Inequalities and Applications 07 07:76 Page of 5 sequently establish three theorems as conclusions in Section. First, the sharp maximal estimates of multilinear iterated commutatorsgenerated by MO functions and multilinear singular integral operators with generalized kernels will be established in heorem.. hen, the boundedness of this kind of multilinear iterated commutators on the product of weighted Lebesgue spaces and the product of variable exponent Lebesgue spaces will be put forward by heorems. and.3, respectively. In addition there are some necessary lemmas in Section 3. he proof of the main results will be given in Section 4. Let us recall some necessary definitions and notations firstly before starting our main results. Definition. [6] Let m N + and Ky 0, y, y,...,y m be a function away from the diagonal y 0 = y = = y m in R n m+. stands for an m-linear singular integral operator defined by f,...,f m x= R n Kx, y, y,...,y m R n f j y j dy dy m, where f j j =,...,m are smooth functions with compact support, and x / m supp f j. If the kernel K satisfies the following two conditions: C: For some C >0and all y 0, y, y,...,y m R n m+ defined away from the diagonal, Ky0, y, y,...,y m C m k,l=0 y k y l mn ; C: Whenever i =,...,m, C ki are positive constants for any k, k,...,k m N +, Ky 0, y,...,y m km y 0 y 0 y m y 0 < km+ y 0 y 0 k y 0 y 0 y y 0 < k + y 0 y 0 K y 0, y,...,y m q dy dy m q y0 y 0 mn q m i= C ki n q ki, where q, q is a fixed pair of positive numbers satisfying q + q =and<q <,thenwe call an m-linear singular integral operator with generalized kernel. If the kernel K satisfies condition and the following condition: Ky0,...,y j,...,y m K y 0,...,y j,...,y m C y j y j ε m k,l=0 y, 3 k y l mn+ε for some ε >0, y j y j max 0 k m y j y k whenever 0 j m,thenwecall astandard m-linear Calderón-Zygmund kernel. If Ky 0, y,...,y m is defined away from the diagonal y 0 = y = = y m in R n m+,then it is calledan m-linear Calderón-Zygmund kernel of type κ when it satisfies condition
Linand ZhangJournal of Inequalities and Applications 07 07:76 Page 3 of 5 and the following condition: Ky 0,...,y j,...,y m K y 0,...,y j,...,y m C y 0 y + + y 0 y m κ mn y j y j y 0 y + + y 0 y m, 4 where κt is a non-negative and non-decreasing function on R + and y j y j max k m y 0 y k whenever 0 j m. It is obvious that condition 4 becomes condition 3 whenκt =t ε for some ε >0 and condition 4 implies condition by putting C ki = κ k i m, i =,...,m, andany < q <. herefore, the standard m-linear Calderón-Zygmund kernel is a special case of the m-linear Calderón-Zygmund kernel of type κ. And the multilinear singular integral with the kernel of type κ can be taken as a special situation of the multilinear singular integral operator with generalized kernel defined in Definition.. hese facts illustrate that our results obtained in this paper will improve most of the earlier conclusions by weakening the conditions of the kernel. Definition. Let be an m-linear singular integral operator with generalized kernel, b =b,...,b m MO m is a group of locally integrable functions and f =f,...,f m. hen the m-linear iterated commutator generated by and b is defined to be b f,...,f m = [ b, [ b,..., [ ] ] b m,[b m, ] m m ] f. If is connected in the usual way to the kernel K studied in this paper, then we can write b f,...,f m x = bj x b j y j Kx, y,...,y m f y f m y m dy dy m. R n m We also denote j b j f x=b j xf,...,f m x f,...,f j, b j f j, f j+,...,f m x. Definition.3 ake positive integers j and m satisfying j m,andletcj m be a family of all finite subsets σ = {σ,...,σj} of {,...,m} with j different elements. If k < l, then σ k<σl. For any σ Cj m,letσ = {,...,m}\σ be the complementary sequence. In particular, C0 m =. For an m-tuple b and σ Cj m,thej-tuple b σ =b σ,...,b σ j isafinite subset of b =b,...,b m. Let be an m-linear singular integral operator with generalized kernel, σ Cj m, b σ = b σ,...,b σ j MO j, the iterated commutator is given by b σ f,...,f m = [ b σ, [ b σ,..., [ ] ] b σ j,[b σ j, ] σ j σ j ]σ σ f.
Linand ZhangJournal of Inequalities and Applications 07 07:76 Page 4 of 5 It can also be written as b σ f,...,f m x = R n m j bσ i x b σ i y σ i Kx, y,...,y m f y f m y m d y, i= where d y = dy dy m.obviously, b σ = b when σ = {,,...,m},and b σ = j b j when σ = {j}. Definition.4 he Hardy-Littlewood maximal operator M is given by Mf x=sup f y dy, x where the supremum is taken over all balls which contain x. We define the operator M s f by M s f =M f s /s,0<s <. he sharp maximal operator M is defined by M f x=sup f y f dy sup x x inf a C f y a dy. Denote the l-sharp maximal operator by M l f =M f l /l,0<l <. Definition.5 [8] Let ω be a non-negative measurable function. For < p <, ω A p if there exists a positive constant C independent of every Q in R n such that p ωx dx ωx dx p, Q Q Q Q where Q represents a cube with the side parallel to the coordinate axes, /p +/p =. When p =,ω belongs to A, if there exists a constant C >0,andforanycubeQ such that ωy dy ωx, a.e. x Q. Q Q Definition.6 Let p :R n [, be a measurable function. Define the variable exponent Lebesgue space L p R n by L p R n { = f measurable : R n y associating with the norm f L p R n {λ = inf >0: R n f x f x the set L p R n comes to be a anach space. λ λ px dx < for some constant λ >0}. px dx },
Linand ZhangJournal of Inequalities and Applications 07 07:76 Page 5 of 5 Definition.7 Let PR n represent the set of measurable functions p :R n [, satisfying <p := essinf px and p x R n + := esssup px<, x R n and R n represent the set of all p PR n for which the Hardy-Littlewood maximal operator M is bounded on L p R n. Main results In this part, we will give the main results in this paper. heorem. Let m, be an m-linear singular integral operator with generalized kernel defined by Definition. and k i = k ic ki <, i =,...,m. Assume for fixed r,...,r m q with /r =/r + +/r m that is bounded from L r L r m into L r,. If b MO m,0< < m, < ε < and q < s <, then there is a constant C >0such that M b f x m b j MO M s f k x+m ε f x k= m σ Cj m j b σ i MO M ε b σ f x i= for all m-tuples f =f,...,f m of bounded measurable functions with compact support. heorem. Let m, be an m-linear singular integral operator with generalized kernel defined by Definition. and k i = k ic ki <, i =,...,m. Assume for fixed r,...,r m q with /r =/r + +/r m that is bounded from L r L r m into L r,. If b MO m, then for any q < p,...,p m < with /p =/p + +/p m, b is bounded from L p w L p m w m into L p w, where w,...,w m A p /q,...,a p m /q and w = m p w p j j. heorem.3 Let m, p, p,...,p m R n with /p =/p + +/p m. Let q j 0 begivenasinlemma3.8 for p j, j =,...,m. is an m-linear singular integral operator with generalized kernel defined by Definition.,<q < min j m q j 0 and k i = k ic ki <, i =,...,m. Assume for fixed r,...,r m q with /r =/r + +/r m that is bounded from L r L r m into L r,. If b MO m, then b is bounded from Lp R n L pm R n into L p R n. 3 Preliminaries Next,wegivesomerequisitelemmas. Lemma 3. [9, 0] Suppose 0<p < q <, then there exists a constant C = C p,q >0such that for any measurable function f, Q /p f L p Q Q /q f L q, Q.
Linand ZhangJournal of Inequalities and Applications 07 07:76 Page 6 of 5 Lemma 3. [] Let f MOR n, p <, r >0,r >0and x R n. hen x, r x,r /p f y fx,r p dy + ln r f MO, where C is a positive constant independent of f, x, r and r. r Lemma 3.3 [0, ] Suppose 0<p, < and w A. here is a positive constant C depending on the A constant of w such that [ M f x ] p [ wx dx M R n R n f x] p wx dx for every function f such that the left-hand side is finite. Lemma 3.4 [3] Let w,...,w m A p,...,a pm with p,...,p m <, and 0< θ,...,θ m <satisfying θ + + θ m =,then w θ wθ m m A max{p,...,p m }. Lemma 3.5 [6] Let m, be an m-linear singular integral operator with generalized kernel defined by Definition. and k i = C k i <, i =,...,m. Suppose for fixed r,...,r m q with /r =/r + +/r m that is bounded from L r L r m into L r,. If 0< </m, then M f x M q f j x for all m-tuples f =f,...,f m of bounded measurable functions with compact support. Lemma 3.6 [8] For ω A p,<p <, there are ω A r for all r > pandω A q for some <q < p. Lemma 3.7 [6] Let m, be an m-linear singular integral operator with generalized kernel defined by Definition. and k i = k ic ki <, i =,...,m. Suppose for fixed r,...,r m q with /r =/r + +/r m that is bounded from L r L r m into L r,. If b j MO for j =,...,m,0< < m, < ε < and q < s <, then M j b j f x b j MO M ε f x+ M s f i x i= for all m-tuples f =f,...,f m of bounded measurable functions with compact support. heaboveresultcanbeobtainedfromtheproofofheorem. in [6]. Lemma 3.8 [4] Suppose p PR n. M is bounded on L p R n if and only if for some <q 0 <, M q0 is bounded on L p R n, where M q0 f =[M f q 0] /q 0.
Linand ZhangJournal of Inequalities and Applications 07 07:76 Page 7 of 5 Lemma 3.9 [3] Let p, p,...,p m PR n satisfying /px=/p x+ +/p m x. For any f j L pj R n, j =,...,m, there is f j L m m f j p L j. R n p R n Lemma 3.0 [5] Given a family of ordered pairs of measurable functions F, for some fixed 0<p 0 <, any f, g F and any w A, f x p 0 wx dx 0 gx p 0 wx dx. R n R n Suppose p PR n satisfying p 0 p. If p p 0 R n, then there exists a constant C >0 such that for any f, g F, f L p R n g L p R n. Lemma 3. [4] If p PR n, then the following four conditions are equivalent. p R n. p R n. 3 For some <p 0 < p, p p 0 R n. 4 For some <p 0 < p, p p 0 R n. Lemma 3. [5] For p PR n, C 0 Rn is dense in L p R n. 4 Proof of the main results We will give the proof of the three theorems in the following article. Proof of heorem. For the sake of simplicity, we only consider the case m =.heproof of other cases is similar. Let f, f be bounded measurable functions with compact support, b, b MO.hen, for any constant λ and λ, b f x= b x λ b x λ f, f x b x λ f,b λ f x b x λ b λ f, f x + b λ f,b λ f x = b x λ b x λ f, f x+ b x λ b λ f, f x+ b x λ b λ f, f x + b λ f,b λ f x. Let C 0 be a constant determined later. Fixed x R n, for any ball x 0, r containing x and 0 < <,wehave b f z C 0 b f z C 0
Linand ZhangJournal of Inequalities and Applications 07 07:76 Page 8 of 5 b z λ b z λ f, f z := I + II + III + IV. b z λ b λ f, f z b z λ b λ f, f z hen we analyze each part separately. Let =6 and λ j =b j = 6 b λ f,b λ f z C0 6 b jx dx, j =,.Since0< < and 0 < < ε <, there exists l such that < l < min{ ε, }.henl < ε and l >.Chooseq, q, such that q + q = l,then q + q + l =andq >,q >. y Hölder s inequality and Lemma 3.,wehave I b z λ b z λ f, f z b z λ q f, f z l l q b MO b MO M l f, f x b MO b MO M ε f, f x. It follows from Hölder s inequality that q b z λ q II b z λ l l b λ f, f z l l Similarly, b MO M l b λ f, f x b MO M ε b λ f, f x = C b MO M ε b f, f x. III b MO M ε b f, f x. In the next, we analyze IV.Wesplitf i into two parts, f i = f 0 i + fi,wherefi 0 = f χ and f i = f i f 0 i, i =,.Choosez 0 3\ and C 0 = b λ f,b λ f z 0, then IV b λ f 0,b λ f 0 z b λ f 0,b λ f z
Linand ZhangJournal of Inequalities and Applications 07 07:76 Page 9 of 5 b λ f b λ f := IV + IV + IV 3 + IV 4.,b λ f 0 z,b λ f z C0 hen we estimate each part respectively. It follows from 0 < < r < and Lemma 3. that IV b λ f 0,b λ f 0 L r b λ f 0,b λ f 0 L r,. Notice that is bounded from L r L r into L r,, r, r q with /r =/r +/r. Let t = s q.sinceq < s <,then<t < and r i t s, i =,.hus, IV 6 6 6 6 6 b y λ r f y r r dy 6 6 b y λ r f y r r dy b y λ r t r t dy 6 6 b MO b MO 6 f y s s dy 6 6 b y λ r t r t dy 6 6 b MO b MO M s f xm s f x. f y s dy s 6 f y r r t t dy 6 f y r r t t dy Since y 6, y 6 c, x 0 and z, then by Lemma 3., IV Kz, y, y b y λ f y b y λ 6 c 6 f y dy dy b y λ f y dy 6 c 6 b y λ f y dy z y n b y λ f y dy 6 k=4 k+ \ k b y λ f y x 0 y n dy
Linand ZhangJournal of Inequalities and Applications 07 07:76 Page 0 of 5 6 k+ 6 k=4 6 b y λ f y dy kn k+ b y λ f y dy 6 k+ k=4 q b y λ q dy 6 kn k+ k+ k+ 6 b y λ q dy q f y q q dy b MO b MO M q f xm q f x b MO b MO M s f xm s f x. kn k k=4 f y q q dy Similarly, IV 3 b MO b MO M s f xm s f x. Since z 0 3\, z, y 6 c, y 6 c,then y z 0 z z 0 and y z 0 z z 0. IV 4 Kz, y, y Kz 0, y, y b y λ 6 c 6 c f y b y λ f y dy dy k = k = k z z 0 y z 0 < k + z z 0 Kz, y, y Kz 0, y, y b y λ k z z 0 y z 0 < k + z z 0 f y b y λ f y dy dy k = k = k z z 0 y z 0 < k + z z 0 b y λ f y Kz, y, y Kz 0, y, y q q dy k z z 0 y z 0 < k + z z 0 b y λ q f y q q dy dy k +4 k = k = k +4 b y λ q f y q q dy
Linand ZhangJournal of Inequalities and Applications 07 07:76 Page of 5 hus, b y λ q f y q q dy k z z 0 y z 0 < k + z z 0 Kz, y, y k z z 0 y z 0 < k + z z 0 Kz 0, y, y q dy dy q k = k = k +4 k z z 0 y z 0 < k + z z 0 k +4 b y λ q f y q q dy b y k λ q f y q q dy k +4 q +4 k +4 k +4 q z z 0 n q C k n q k C k n q k k = k = k +4 k +4 k +4 k +4 k +4 k +4 f y q t q t dy b y λ q t q dy t b y λ q t q dy t f y k q t q t dy Ck C k +4 k +4 b MO b MO M s f xm s f x k C k k C k b MO b MO M s f xm s f x. k = k = IV b MO b MO M s f xm s f x. herefore, M b f x = M b f x sup b x f z C 0 b MO b MO Ms f xm s f x+m ε f, f x b MO M ε b f, f x+ b MO M ε b f, f x, which completes the proof of heorem..
Linand ZhangJournal of Inequalities and Applications 07 07:76 Page of 5 Proof of heorem. We have w A p max{ q,..., pm qm } A from Lemma 3.4. y Lemma 3.6, for every i =,...,m, sincew i A pi /q,thereexistsl i satisfying < l i < p i /q and w i A li. Since q < p i /l i,thereexistss i satisfying q < s i < p i /l i < p i.lets = min i m s i,thenwehave q < s < p i.sincel i < p i /s i p i /s,thenw i A li A pi /s, i =,...,m. Choose, ε, ε,...,ε m satisfying 0 < < ε < ε < < ε m <. It follows from Lemma 3.3 m and Lemma 3.5 that Mεj f L p w M εj f m L p w M q f i, j =,...,m. y heorem.,wehave k= i= L p w M b f L p w m b j MO M s f k + Mε f L p w L p w m σ Cj m j b σ i Mε MO b f σ L p w i= m b j MO M s f k + Mε f L p w L p w k= m σ Cj m j b σ i MO M ε b f σ L p w. i= In order to reduce the dimension of MO functions in the commutators, we apply heorem. again to M ε b σ f L p w. Let σ = {σ,...,σ j} and σ = {σ j +,...,σ m}, A h = {σ :anyfinitesubsetofσ with different elements} and σ = σ σ. It follows from heorem. that M ε b f σ L p w m b σ k MO M s f l + Mε f L p w L p w k=j+ m j h= σ A h i= l= h b σ i MO M ε b f σ L p w. y putting the formula above into M b f L p w, we can reduce the dimension of MO functions. Repeating the process above and using Lemma 3.7,wecanget M b f L p w b j MO A m+ m, n M s f k k= L p w
Linand ZhangJournal of Inequalities and Applications 07 07:76 Page 3 of 5 + A m, n Mε f L p w + A m, n Mε f L p w + + A m m, n Mεm f L p, w where A m, n, A m, n,...,a m+ m, n are finite real numbers related to m and n. hen, by Lemma 3.3 and Lemma 3.5, b f L p w M b f L p w M b f L p w b j MO A m+ m, n M s f k = C = C k= L p w + A m, n Mε f L p w + A m, n Mε f L p w + + A m m, n Mεm f L p w b j MO A m+ m, n M s f k + A m+ m, n M q f j m b j MO M s f j b j MO b j MO b j MO b j MO m m m m k= L p w L p w Ms f j p L j w j M fj s s f j s s L L p j s w j f j L p j w j, p j s w j L p w which completes the proof of heorem.. Proof of heorem.3 Let q 0 = min j m q j 0,thenq < q 0 <. It follows from p R n and Lemma 3. that there exists p 0 satisfying < p 0 < p and p p 0 R n. Choose, ε, ε,...,ε m satisfying 0 < < ε < ε < < ε m < m. For any w A,wehave R n b f x p0 wx dx = b f p 0 L p 0 w M b f p 0 L p 0 w
Linand ZhangJournal of Inequalities and Applications 07 07:76 Page 4 of 5 M b f p 0 L p 0 w m b j MO A m+ m, n M q0 f j L p 0 w + A m, n M ε f L p 0 w + A m, n M ε f L p 0 w + + A m m, n M εm f L p 0 w p0 m b j MO A m+ m, n M q0 f j p0 + A m+ m, n M q f j L p 0 w m m p0 b j MO f j m p0 = C b j MO M q j 0 R n m L p 0 w L p 0 w p0 M j q f j x wx dx 0 for all f =f,...,f m which are bounded measurable functions with compact support, where A m, n, A m, n,...,a m+ m, n are finite real numbers related to m and n. Applying Lemma 3.0 to b f, m M q0f j j, we have b f m L p R n M q j 0 f j L p R n hen it follows from Lemma 3.9 and Lemma 3.8 that b f m L p R n M j q f j 0 L p j m f R n j p L j, R n. which completes the proof of heorem.3. Acknowledgements his work was supported by the National Natural Science Foundation of China 67397, the Fundamental Research Funds for the Central Universities 009QS6, and the Yue Qi Young Scholar of China University of Mining and echnology eijing. Competing interests he authors declare that they have no competing interests. Authors contributions he authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript. Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: September 07 Accepted: 5 October 07
Linand ZhangJournal of Inequalities and Applications 07 07:76 Page 5 of 5 References. Kenig, C, Stein, E: Multilinear estimates and fractional integration. Math. Res. Lett. 6, -5 999. Grafakos, L, orres, RH: Multilinear Calderón-Zygmund theory. Adv. Math. 65, 4-64 00 3. Lu, GZ, Zhang, P: Multilinear Calderón-Zygmund operators with kernels of Dini s type and applications. Nonlinear Anal. MA 07, 9-7 04 4. Pérez, C, rujillo-gonzález, R: Sharp weighted estimates for multilinear commutators. J. Lond. Math. Soc. 65, 67-69 00 5. Si, ZY, Lu, SZ: Weighted estimates for iterated commutators of multilinear fractional operators. Acta Math. Sin. 8, 769-778 0 6. Lin, Y, Xiao, YY: Multilinear singular integral operators with generalized kernels and their multilinear commutators. Acta Math. Sin. 33, 443-46 07 7. Pérez, C, Pradolini, G, orres, RH, rujillo-gonzález, R: Endpoint estimates for iterated commutators of multilinear singular integral. ull. Lond. Math. Soc. 46, 6-4 04 8. Muckenhoupt, : Weighted norm inequalities for the Hardy maximal function. rans. Am. Math. Soc. 65, 07-6 97 9. Garca-Cuerva, J, Rubio de Francia, JL: Weighted Norm Inequalities and Related opics. North-Holland Math. Studies, vol. 6. North-Holland, Amsterdam 985 0. Lerner, AK, Ombrosi, S, orres, RH, Pérez, C, rujillo-gonzález, R: New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory. Adv. Math. 0, -64 009. Lin, Y, Lu, SZ: Strongly singular Calderón-Zygmund operators and their commutators. Jordan J. Math. Stat., 3-49 008. Fefferman, C, Stein, EM: H p spaces of several variables. Acta Math. 9, 37-93 97 3. Grafakos, L, Martell, JM: Extrapolation of weighted norm inequalities for multivariable operators and applications. J. Geom. Anal. 4, 9-46 004 4. Diening, L: Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces. ull. Sci. Math. 9, 657-700 005 5. Diening, L, Harjulehto, P, Hästö, P, R užička, M: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Math. Springer, erlin 0