In this paper, we shall deal with the boundedness of the Littlewood-Paley operators with rough kernel. We prove the boundedness of the Lusin-area integral μΩs and Littlewood-Paley functions μΩ and μλ^* on the w...In this paper, we shall deal with the boundedness of the Littlewood-Paley operators with rough kernel. We prove the boundedness of the Lusin-area integral μΩs and Littlewood-Paley functions μΩ and μλ^* on the weighted amalgam spaces (Lω^q,L^p)^α(R^n)as 1〈q≤α〈p≤∞.展开更多
In this paper, for the multilinear oscillatory singular integral operators TA1,A2,...Ar defined by TA1,A2,...,Arf(x) = p.v.∫R^n ^e^iP(x,y)Ω(x - y)/|x - y|^n+M r∏s=1 Rms+1(As;x,y)f(y)dy, n≥2 where P...In this paper, for the multilinear oscillatory singular integral operators TA1,A2,...Ar defined by TA1,A2,...,Arf(x) = p.v.∫R^n ^e^iP(x,y)Ω(x - y)/|x - y|^n+M r∏s=1 Rms+1(As;x,y)f(y)dy, n≥2 where P(x,y) is a nontrivial and real-valued polynomial defined on R^n×R^n,Ω(x) is homogeneous of degree zero on R^n, As(x) has derivatives of order ms in ∧βs (0〈βs〈 1), Rms+1 (As;x, y) denotes the (ms+1)-st remainder of the Taylor series of As at x expended about y (s = 1, 2, ..., r), M = ∑s^r =1 ms, the author proves that if 0 〈=β1=∑s^r=1 βs〈1,and Ω∈L^q(S^n-1) for some q 〉 1/(1 -β), then for any p∈(1, ∞), and some appropriate 0 〈β〈 1, TA1,A2,...,Ar, is bounded on L^P(R^n).展开更多
This paper studies the mixed radial-angular integrability of parametric Marcinkiewicz integrals along"polynomial curves".Under the assumption that the kernels satisfy certain rather weak size conditions on t...This paper studies the mixed radial-angular integrability of parametric Marcinkiewicz integrals along"polynomial curves".Under the assumption that the kernels satisfy certain rather weak size conditions on the unit sphere with radial roughness,the authors prove that such operators are bounded on the mixed radial-angular spaces.Meanwhile,corresponding vector-valued versions are also obtained.展开更多
Let n≥2. In this paper, the author establishes the L2 (Rx)-boundedness of some oscillatory singular integrals with variable rough kernels by means of some estimates on hyper geometric functions and confluent hyper ge...Let n≥2. In this paper, the author establishes the L2 (Rx)-boundedness of some oscillatory singular integrals with variable rough kernels by means of some estimates on hyper geometric functions and confluent hyper geometric funtions.展开更多
In this article, we consider a fast algorithm for first generation Calderon-Zygmund operators. First, we estimate the convergence speed of the relative approximation algorithm. Then, we establish the continuity on Bes...In this article, we consider a fast algorithm for first generation Calderon-Zygmund operators. First, we estimate the convergence speed of the relative approximation algorithm. Then, we establish the continuity on Besov spaces and Triebel-Lizorkin spaces for the oper- ators with rough kernel.展开更多
In this paper, we will prove the Triebel-Lizorkin boundedness for some oscillatory singular integrals with the kernel (x) satisfying a condition introduced by Grafakos and Stefanov. Our theorems will be proved under...In this paper, we will prove the Triebel-Lizorkin boundedness for some oscillatory singular integrals with the kernel (x) satisfying a condition introduced by Grafakos and Stefanov. Our theorems will be proved under various conditions on the phase function, radial and nonradial. Since the L p boundedness of these operators is not complete yet, the theorems extend many known results.展开更多
The Fourier transform and the Littlewood-Paley theory are used to give the weighted boundedness of a strongly singular integral operator defined in this paper. The paper shows that the strongly singular integral opera...The Fourier transform and the Littlewood-Paley theory are used to give the weighted boundedness of a strongly singular integral operator defined in this paper. The paper shows that the strongly singular integral operator is bounded from the Sobolev space to the Lebesgue space.展开更多
This paper is concerning the commutators generated by the multilinear singular integral with rough kernels and BMO functions. The boundedness of the multilinear commutators Tb→ (f→) is established on the Morrey-He...This paper is concerning the commutators generated by the multilinear singular integral with rough kernels and BMO functions. The boundedness of the multilinear commutators Tb→ (f→) is established on the Morrey-Herz space by using the John-Nirenberg inequality.展开更多
In this paper,the boundedness of the multilinear operators with rough kernel on the Herz-type spaces is discussed.It is proved that M A,Ω,T A,Ω are bounded on α,p q(Rn) while Mλ A,Ω,TL A,Ω are bou...In this paper,the boundedness of the multilinear operators with rough kernel on the Herz-type spaces is discussed.It is proved that M A,Ω,T A,Ω are bounded on α,p q(Rn) while Mλ A,Ω,TL A,Ω are bounded from α,p 1 q 1(Rn) to α,p 2 q 2(Rn),respectively.展开更多
The authors establish λ-central BMO estimates for commutators of singular integral operators with rough kernels on central Morrey spaces. Moreover, the boundedness of a class of multisublinear operators on the produc...The authors establish λ-central BMO estimates for commutators of singular integral operators with rough kernels on central Morrey spaces. Moreover, the boundedness of a class of multisublinear operators on the product of central Morrey spaces is discussed. As its special cases, the corresponding results of multilinear Calderon-Zygmund operators and multilinear fractional integral operators can be deduced, resDectivelv.展开更多
For a class of multilinear singular integral operators TA,$$T_A f\left( x \right) = \int {_{\Ropf^n} } {{\Omega \left( {x - y} \right)} \over {\left| {x - y} \right|^{n + m - 1} }}R_m \left( {A;x,y} \right)f\left( y \...For a class of multilinear singular integral operators TA,$$T_A f\left( x \right) = \int {_{\Ropf^n} } {{\Omega \left( {x - y} \right)} \over {\left| {x - y} \right|^{n + m - 1} }}R_m \left( {A;x,y} \right)f\left( y \right)dy,$$where Rm (A; x, y) denotes the m-th Taylor series remainder of A at x expanded about y, A has derivatives of order m m 1 in $\dot \Lambda_\beta $(0 < # < 1), OHgr;(x) ] L^s(S^nm1)($s \ge {n \over {n - \beta }}$) is homogeneous of degree zero, the authors prove that TA is bounded from L^p(A^n) to L^q) (A^n) (${1 \over p} - {1 \over q} = {\beta \over n},\,1 < p < {n \over \beta }$) and from L^1 (A^n) to L^n/(nm#), ^X (A^n) with the bound $C\sum\nolimits_{\left| \gamma \right| = m - 1} {} \left\|\left\| {D^\gamma A} \right\|\right\|_{\dot \Lambda_\beta} $. And if Q has vanishing moments of order m m 1 and satisfies some kinds of Dini regularity otherwise, then TA is also bounded from L^p (A^n) to ${\dot F}^{\beta,\infty}_p$ (A^n)(1 < s' < p < X) with the bound $C\sum\nolimits_{\left| \gamma \right| = m - 1} {} \left\| \left\|{D^\gamma A} \right\|\right\|_{\dot \Lambda _\beta } $.展开更多
This paper is devoted to the study of a class of singular integral operators defined by polynomial mappings on product domains. Some rather weak size conditions, which imply the Lp boundedness of these singular integr...This paper is devoted to the study of a class of singular integral operators defined by polynomial mappings on product domains. Some rather weak size conditions, which imply the Lp boundedness of these singular integral operators as well as the corresponding maximal truncated singular integral operators for some fixed 1〈p〈 ∞,are given.展开更多
The Lp bounds for the parametric Marcinkiewicz integrals associated to compound mapq pings, which contain many classical surfaces as model examples, are given, where the kernels of our operators are rather rough on th...The Lp bounds for the parametric Marcinkiewicz integrals associated to compound mapq pings, which contain many classical surfaces as model examples, are given, where the kernels of our operators are rather rough on the unit sphere as well as in the radial direction. These results substan- tially improve and extend some known results.展开更多
Let L =-△+V(x) be a Schrodinger operator, where △ is the Laplacian on R^n,while nonnegative potential V(x) belonging to the reverse Holder class. The aim of this paper is to give generalized weighted Morrey estimate...Let L =-△+V(x) be a Schrodinger operator, where △ is the Laplacian on R^n,while nonnegative potential V(x) belonging to the reverse Holder class. The aim of this paper is to give generalized weighted Morrey estimates for the boundedness of Marcinkiewicz integrals with rough kernel associated with Schrodinger operator and their commutators.Moreover, the boundedness of the commutator operators formed by BMO functions and Marcinkiewicz integrals with rough kernel associated with Schrodinger operators is discussed on the generalized weighted Morrey spaces. As its special cases, the corresponding results of Marcinkiewicz integrals with rough kernel associated with Schrodinger operator and their commutators have been deduced, respectively. Also, Marcinkiewicz integral operators, rough Hardy-Littlewood(H-L for short) maximal operators, Bochner-Riesz means and parametric Marcinkiewicz integral operators which satisfy the conditions of our main results can be considered as some examples.展开更多
This paper is primarily concerned with the proof of the Lp boundedness of parabolic Littlewood-Paley operators with kernels belonging to certain block spaces. Some known results are essentially extended and improved.
This paper is concerned with singular integral operators on product domains with rough kernels both along radial direction and on spherical surface. Some rather weaker size conditions, which imply the L^P-boundedness ...This paper is concerned with singular integral operators on product domains with rough kernels both along radial direction and on spherical surface. Some rather weaker size conditions, which imply the L^P-boundedness of such operators for certain fixed p (1 〈 p 〈 ∞), are given.展开更多
In this paper, the authors give the boundedness of the commutator [b,μΩ,γ] from the homogeneous Sobolev space LP(R^n) to the Lebesgue space L^p(R^n) for 1 〈 p 〈 ∞, where μΩ,γ denotes the Marcinkiewicz int...In this paper, the authors give the boundedness of the commutator [b,μΩ,γ] from the homogeneous Sobolev space LP(R^n) to the Lebesgue space L^p(R^n) for 1 〈 p 〈 ∞, where μΩ,γ denotes the Marcinkiewicz integral with rough hypersingular kernel defined by μΩ,γf(x)=(∫0^∞|∫|x-y|≤tΩ(x-y)/|x-y|^n-1f(y)dy|^2dt/t^3+2γ)^1/2,with Ω∈L^1(S^n-1)for 0〈γ〈min(n/2,n/p)or Ω∈L(log+L)^β(S^n-1)for|1-2/p|〈β〈1(0〈γ〈n/2),respectively.展开更多
In this paper, we study the generalized Marcinkiewicz integral operators MΩ,r on the product space Rn ×Rm. Under the condition that Ω is a function in certain block spaces, which is optimal in some senses, the ...In this paper, we study the generalized Marcinkiewicz integral operators MΩ,r on the product space Rn ×Rm. Under the condition that Ω is a function in certain block spaces, which is optimal in some senses, the boundedness of such operators from Triebel-Lizorkin spaces to Lp spaces is obtained.展开更多
In this paper,we prove that the commutators of maximal hypersingular integrals with rough kernels are bounded from the Sobolev space Lpγ(Rn) to the Lebesgue space Lp(Rn),which is a substantial improvement and an exte...In this paper,we prove that the commutators of maximal hypersingular integrals with rough kernels are bounded from the Sobolev space Lpγ(Rn) to the Lebesgue space Lp(Rn),which is a substantial improvement and an extension of some known results.展开更多
基金supported in part by National Natural Foundation of China (Grant No. 11161042 and No. 11071250)
文摘In this paper, we shall deal with the boundedness of the Littlewood-Paley operators with rough kernel. We prove the boundedness of the Lusin-area integral μΩs and Littlewood-Paley functions μΩ and μλ^* on the weighted amalgam spaces (Lω^q,L^p)^α(R^n)as 1〈q≤α〈p≤∞.
文摘In this paper, for the multilinear oscillatory singular integral operators TA1,A2,...Ar defined by TA1,A2,...,Arf(x) = p.v.∫R^n ^e^iP(x,y)Ω(x - y)/|x - y|^n+M r∏s=1 Rms+1(As;x,y)f(y)dy, n≥2 where P(x,y) is a nontrivial and real-valued polynomial defined on R^n×R^n,Ω(x) is homogeneous of degree zero on R^n, As(x) has derivatives of order ms in ∧βs (0〈βs〈 1), Rms+1 (As;x, y) denotes the (ms+1)-st remainder of the Taylor series of As at x expended about y (s = 1, 2, ..., r), M = ∑s^r =1 ms, the author proves that if 0 〈=β1=∑s^r=1 βs〈1,and Ω∈L^q(S^n-1) for some q 〉 1/(1 -β), then for any p∈(1, ∞), and some appropriate 0 〈β〈 1, TA1,A2,...,Ar, is bounded on L^P(R^n).
基金supported by the NSFC(11771358,11701333,11871101)。
文摘This paper studies the mixed radial-angular integrability of parametric Marcinkiewicz integrals along"polynomial curves".Under the assumption that the kernels satisfy certain rather weak size conditions on the unit sphere with radial roughness,the authors prove that such operators are bounded on the mixed radial-angular spaces.Meanwhile,corresponding vector-valued versions are also obtained.
基金Dachun Yang was supported by the Croucher Foundation Chinese Visitorships 1999-2000 of Hong Kong and me NNSF(19131080)of China
文摘Let n≥2. In this paper, the author establishes the L2 (Rx)-boundedness of some oscillatory singular integrals with variable rough kernels by means of some estimates on hyper geometric functions and confluent hyper geometric funtions.
基金Supported by NNSF of China(11271209,1137105,11571261)and SRFDP(20130003110003)
文摘In this article, we consider a fast algorithm for first generation Calderon-Zygmund operators. First, we estimate the convergence speed of the relative approximation algorithm. Then, we establish the continuity on Besov spaces and Triebel-Lizorkin spaces for the oper- ators with rough kernel.
基金Supported by the National Natural Science Foundation of China (11026104, 11201103, 11226108)
文摘In this paper, we will prove the Triebel-Lizorkin boundedness for some oscillatory singular integrals with the kernel (x) satisfying a condition introduced by Grafakos and Stefanov. Our theorems will be proved under various conditions on the phase function, radial and nonradial. Since the L p boundedness of these operators is not complete yet, the theorems extend many known results.
基金Project supported by the National Natural Science Foundation of China (No. 10771110)the Major Project of the Ministry of Education of China (No. 309018)
文摘The Fourier transform and the Littlewood-Paley theory are used to give the weighted boundedness of a strongly singular integral operator defined in this paper. The paper shows that the strongly singular integral operator is bounded from the Sobolev space to the Lebesgue space.
基金supported by the China National Natural Since Foundation (11161042, 11071250)
文摘This paper is concerning the commutators generated by the multilinear singular integral with rough kernels and BMO functions. The boundedness of the multilinear commutators Tb→ (f→) is established on the Morrey-Herz space by using the John-Nirenberg inequality.
基金Supported by the National Natural Science Foundation of China( 1 9631 0 80,1 9971 0 1 0 ) ,the973terms:( 1 9990 75 1 0 5 ) and the Natural Science Foundation of the Zhejiang Province( RC971 0 7)
文摘In this paper,the boundedness of the multilinear operators with rough kernel on the Herz-type spaces is discussed.It is proved that M A,Ω,T A,Ω are bounded on α,p q(Rn) while Mλ A,Ω,TL A,Ω are bounded from α,p 1 q 1(Rn) to α,p 2 q 2(Rn),respectively.
基金National Natural Science Foundation of China (10571014)the Doctoral Programme Foundation of Institution of Higher Education of China (20040027001)
文摘The authors establish λ-central BMO estimates for commutators of singular integral operators with rough kernels on central Morrey spaces. Moreover, the boundedness of a class of multisublinear operators on the product of central Morrey spaces is discussed. As its special cases, the corresponding results of multilinear Calderon-Zygmund operators and multilinear fractional integral operators can be deduced, resDectivelv.
文摘For a class of multilinear singular integral operators TA,$$T_A f\left( x \right) = \int {_{\Ropf^n} } {{\Omega \left( {x - y} \right)} \over {\left| {x - y} \right|^{n + m - 1} }}R_m \left( {A;x,y} \right)f\left( y \right)dy,$$where Rm (A; x, y) denotes the m-th Taylor series remainder of A at x expanded about y, A has derivatives of order m m 1 in $\dot \Lambda_\beta $(0 < # < 1), OHgr;(x) ] L^s(S^nm1)($s \ge {n \over {n - \beta }}$) is homogeneous of degree zero, the authors prove that TA is bounded from L^p(A^n) to L^q) (A^n) (${1 \over p} - {1 \over q} = {\beta \over n},\,1 < p < {n \over \beta }$) and from L^1 (A^n) to L^n/(nm#), ^X (A^n) with the bound $C\sum\nolimits_{\left| \gamma \right| = m - 1} {} \left\|\left\| {D^\gamma A} \right\|\right\|_{\dot \Lambda_\beta} $. And if Q has vanishing moments of order m m 1 and satisfies some kinds of Dini regularity otherwise, then TA is also bounded from L^p (A^n) to ${\dot F}^{\beta,\infty}_p$ (A^n)(1 < s' < p < X) with the bound $C\sum\nolimits_{\left| \gamma \right| = m - 1} {} \left\| \left\|{D^\gamma A} \right\|\right\|_{\dot \Lambda _\beta } $.
文摘This paper is devoted to the study of a class of singular integral operators defined by polynomial mappings on product domains. Some rather weak size conditions, which imply the Lp boundedness of these singular integral operators as well as the corresponding maximal truncated singular integral operators for some fixed 1〈p〈 ∞,are given.
基金Supported by National Natural Science Foundation of China(Grant Nos.11071200,11371295)NSF ofFujian Province of China(Grant No.2010J01013)
文摘The Lp bounds for the parametric Marcinkiewicz integrals associated to compound mapq pings, which contain many classical surfaces as model examples, are given, where the kernels of our operators are rather rough on the unit sphere as well as in the radial direction. These results substan- tially improve and extend some known results.
文摘Let L =-△+V(x) be a Schrodinger operator, where △ is the Laplacian on R^n,while nonnegative potential V(x) belonging to the reverse Holder class. The aim of this paper is to give generalized weighted Morrey estimates for the boundedness of Marcinkiewicz integrals with rough kernel associated with Schrodinger operator and their commutators.Moreover, the boundedness of the commutator operators formed by BMO functions and Marcinkiewicz integrals with rough kernel associated with Schrodinger operators is discussed on the generalized weighted Morrey spaces. As its special cases, the corresponding results of Marcinkiewicz integrals with rough kernel associated with Schrodinger operator and their commutators have been deduced, respectively. Also, Marcinkiewicz integral operators, rough Hardy-Littlewood(H-L for short) maximal operators, Bochner-Riesz means and parametric Marcinkiewicz integral operators which satisfy the conditions of our main results can be considered as some examples.
基金Supported by National Natural Science Foundation of China (Grant Nos. 10961015, 10871173)National Natural Science Foundation of Jiangxi Province (2008GZS0051) the doctor foundation of Jiangxi Normal University (2443)
文摘This paper is primarily concerned with the proof of the Lp boundedness of parabolic Littlewood-Paley operators with kernels belonging to certain block spaces. Some known results are essentially extended and improved.
基金Supported by National Natural Science Foundation of China (Grant Nos.10771054,10961003,11071200)Natural Science Foundation of Fujian Province of China (Grant No.2010J01013)
文摘This paper is concerned with singular integral operators on product domains with rough kernels both along radial direction and on spherical surface. Some rather weaker size conditions, which imply the L^P-boundedness of such operators for certain fixed p (1 〈 p 〈 ∞), are given.
基金Supported by National Natural Science Foundation of China (Grant Nos. 10931001, 10901017) and Specialized Research Fund for the Doctoral Program of China (Grant No. 20090003110018)Acknowledgements The authors would like to express their gratitude to the referee for his/her very careful reading and many important valuable comments.
文摘In this paper, the authors give the boundedness of the commutator [b,μΩ,γ] from the homogeneous Sobolev space LP(R^n) to the Lebesgue space L^p(R^n) for 1 〈 p 〈 ∞, where μΩ,γ denotes the Marcinkiewicz integral with rough hypersingular kernel defined by μΩ,γf(x)=(∫0^∞|∫|x-y|≤tΩ(x-y)/|x-y|^n-1f(y)dy|^2dt/t^3+2γ)^1/2,with Ω∈L^1(S^n-1)for 0〈γ〈min(n/2,n/p)or Ω∈L(log+L)^β(S^n-1)for|1-2/p|〈β〈1(0〈γ〈n/2),respectively.
基金Supported by National Natural Science Foundation of China (Grant No. G11071200)Natural Science Foundation of Fujian Province of China (Grant No. 2010J01013)
文摘In this paper, we study the generalized Marcinkiewicz integral operators MΩ,r on the product space Rn ×Rm. Under the condition that Ω is a function in certain block spaces, which is optimal in some senses, the boundedness of such operators from Triebel-Lizorkin spaces to Lp spaces is obtained.
基金supported by National Natural Science Foundation of China (Grant Nos.10901017 and 10931001)Program for New Century Excellent Talents in University (Grant No. NCET-11-0574)Doctoral Fund of Ministry of Education of China (Grant No. 20090003110018)
文摘In this paper,we prove that the commutators of maximal hypersingular integrals with rough kernels are bounded from the Sobolev space Lpγ(Rn) to the Lebesgue space Lp(Rn),which is a substantial improvement and an extension of some known results.