In this paper,we consider the boundedness on Triebel-Lizorkin spaces for the d-dimensional Calder´on commutator defined by TΩ,af(x)=p.v.∫R_(d)Ω(x−y)/|x−y|^(d+1)(a(x)−a(y))f(y)dy,where Ω is homogeneous of degr...In this paper,we consider the boundedness on Triebel-Lizorkin spaces for the d-dimensional Calder´on commutator defined by TΩ,af(x)=p.v.∫R_(d)Ω(x−y)/|x−y|^(d+1)(a(x)−a(y))f(y)dy,where Ω is homogeneous of degree zero,integrable on Sd−1 and has a vanishing moment of order one,and a is a function on Rd such that∇a∈L^(∞)(R^(d)).We prove that if 1<p,q<∞andΩ∈L(log L)^(2 q)(S^(d−1))with q=max{1/q,1/q′},then TΩ,a is bounded on Triebel-Lizorkin spaces˙F_(p)^(0)q(R^(d)).展开更多
This paper is devoted to studying the behaviors of the fractional type Marcinkiewicz integralsμΩ,βand the commutatorsμΩ,βb generated byμΩ,βwith b b∈Lloc(Rn)on weighted Hardy spaces.Under the assumption of th...This paper is devoted to studying the behaviors of the fractional type Marcinkiewicz integralsμΩ,βand the commutatorsμΩ,βb generated byμΩ,βwith b b∈Lloc(Rn)on weighted Hardy spaces.Under the assumption of that the homogeneous kernelΩsatisfies certain regularities,the authors obtain the boundedness ofμΩ,βfrom the weighted Hardy spaces Hωpp(Rn)to the weighted Lebesgue spaces Lωqq(Rn)for n/(n+β)≤<p≤1 with 1/q=1/p-β/n,as well as the same(Hωpp,Lωqq)-boudedness ofμΩ,βb when b belongs to BMOωp,p(Rn),which is a non-trivial subspace of BMO(Rn).展开更多
In this paper,the authors consider theω-type Calder´on-Zygmund operator T_(ω)and the commutator[b,T_(ω)]generated by a symbol function b on the Lorentz space L^(p,r)(X)over the homogeneous space(X,d,μ).The bo...In this paper,the authors consider theω-type Calder´on-Zygmund operator T_(ω)and the commutator[b,T_(ω)]generated by a symbol function b on the Lorentz space L^(p,r)(X)over the homogeneous space(X,d,μ).The boundedness and the compactness of the commutator[b,T_(ω)]on Lorentz space L^(p,r)(X)are founded for any p∈(1,∞)and r∈[1,∞).展开更多
Let p ∈(n/(n + l), 1]. The authors investigate the (Hbp(Rn), Lp(Rn)-type and (Hbp,∞(Rn), Lp,∞ (Rn)-type continuities for the maximal operators associated with the commutators of Bochner-mesz operators with BMO(Rn )...Let p ∈(n/(n + l), 1]. The authors investigate the (Hbp(Rn), Lp(Rn)-type and (Hbp,∞(Rn), Lp,∞ (Rn)-type continuities for the maximal operators associated with the commutators of Bochner-mesz operators with BMO(Rn ) functions, where Hbp(Rn ) and Hpb∞(Rn) are, respectively, the variants of the standard Hardy spaces and the standard weak Hardy spaces.展开更多
In this paper,the authors introduce the central bounded oscillation space CBMO q (R n),let [b,T,α ] be the commutator generated by fractional integral operators with variable kernels and CBMO function,we establish th...In this paper,the authors introduce the central bounded oscillation space CBMO q (R n),let [b,T,α ] be the commutator generated by fractional integral operators with variable kernels and CBMO function,we establish the boundedness of [b,T,α ] on homogeneous Morrey-Herz spaces.展开更多
Let L=-div(A▽) be a second order divergence form elliptic operator, where A is an accretive, n×n matrix with bounded measurable complex coefficients on R^n. Let L^α/2 (0 <α< 1) denotes the fractional dif...Let L=-div(A▽) be a second order divergence form elliptic operator, where A is an accretive, n×n matrix with bounded measurable complex coefficients on R^n. Let L^α/2 (0 <α< 1) denotes the fractional differential operator associated with L and (-△)^α/2b ∈ L^n/α(R^n). In this article, we prove that the commutator[b, L^α/2] is bounded from the homogenous Sobolev space Lα%2 (R^n) to L^2(R^n).展开更多
This paper establishes some strong type and weak type estimates for commutator [b,I1] on Herz-type spaces, where b E BMO(Rn) and I1 is a fractional integration with O < l < n.
Let T be an anisotropic Calderón-Zygmund operator andφ:R^n×[0,∞)→[0,∞)be an anisotropic Musielak-Orlicz function withφ(x,·)being an Orlicz function andφ(·,t)being a Muckenhoupt A∞(A)weight.I...Let T be an anisotropic Calderón-Zygmund operator andφ:R^n×[0,∞)→[0,∞)be an anisotropic Musielak-Orlicz function withφ(x,·)being an Orlicz function andφ(·,t)being a Muckenhoupt A∞(A)weight.In this paper,our goal is to study two boundedness theorems for commutators of anisotropic Calderon-Zygmund operators.Precisely,when b∈BMOw(R^n,A)(a proper subspace of anisotropic bounded mean oscillation space BMO(R^n,A)),the commutator[b,T]is bounded from anisotropic weighted Hardy space H^1ω(R^n,A)to weighted Lebesgue space L^1ω(R^n)and when b∈BMO(R^n)(bounded mean oscillation space),the commutator[b,T]is bounded on Musielak-Orlicz space L^φ(R^n),which are extensions of the isotropic setting.展开更多
In this paper, the behavior for commutators of a class of bilinear singular integral operators associated with non-smooth kernels on the product of weighted Lebesgue spaces is considered. By some new maximal functions...In this paper, the behavior for commutators of a class of bilinear singular integral operators associated with non-smooth kernels on the product of weighted Lebesgue spaces is considered. By some new maximal functions to control the commutators of bilinear singular integral operators and CMO(Rn) functions, compactness for the commutators is proved.展开更多
In this paper, we study the boundedness of the fractional integral operator and their commutator on Herz spaecs with two variable exponents . By using the properties of the variable exponents Lebesgue spaces, the boun...In this paper, we study the boundedness of the fractional integral operator and their commutator on Herz spaecs with two variable exponents . By using the properties of the variable exponents Lebesgue spaces, the boundedness of the fractional integral operator and their commutator generated by Lipschitz function is obtained on those Herz spaces.展开更多
In this paper, the authors establish the(L^p(μ), L^q(μ))-type estimate for fractional commutator generated by fractional integral operators Tα with Lipschitz functions(b ∈ Lipβ(μ)),where 1 < p < 1/(α + β...In this paper, the authors establish the(L^p(μ), L^q(μ))-type estimate for fractional commutator generated by fractional integral operators Tα with Lipschitz functions(b ∈ Lipβ(μ)),where 1 < p < 1/(α + β) and 1/q = 1/p-(α + β), and obtain their weak(L^1(μ), L^(1/(1-α-β))(μ))-type. Moreover, the authors also consider the boundedness in the case that 1/(α+β) < p < 1/α,1/α≤ p ≤∞ and the endpoint cases, namely, p = 1/(α + β).展开更多
Our aim in this paper is to prove the boundedness of commutators of Calderón-Zygmund operator with the Lipschitz function or BOM function on Herz-type Hardy space with variable exponent.
The commutators of oscillatory singular integral operators with homogeneous kernel are studied, where Ω is homogeneous of degree zero, has mean value zero on the unit sphere. It is proved that Ω∈L (logL)(k+1)(S(n-1...The commutators of oscillatory singular integral operators with homogeneous kernel are studied, where Ω is homogeneous of degree zero, has mean value zero on the unit sphere. It is proved that Ω∈L (logL)(k+1)(S(n-1)) is a sufficient condition under which the k-th order commutator is bounded on L2(Rn).展开更多
Let α∈0,(n-1)/2 and T~α be the Bochner-Riesz operator of order α. In this paper, for n = 2 and n ≥ 3, the compactness on Lebesgue spaces and Morrey spaces are considered for the commutator of Bochner-Riesz operat...Let α∈0,(n-1)/2 and T~α be the Bochner-Riesz operator of order α. In this paper, for n = 2 and n ≥ 3, the compactness on Lebesgue spaces and Morrey spaces are considered for the commutator of Bochner-Riesz operator generated by CMO(R^n) function and T~α.展开更多
Let L be the infinitesimal generator of an analytic semigroup on L^2(R^n)with pointwise upper bounds on heat kernel,and denote by L^(-α/2)the fractional integrals of L.For a BMO function b(x),we show a weak type Llog...Let L be the infinitesimal generator of an analytic semigroup on L^2(R^n)with pointwise upper bounds on heat kernel,and denote by L^(-α/2)the fractional integrals of L.For a BMO function b(x),we show a weak type Llog L estimate of the commutators [b,L^(-α/2)](f)(x) = b(x)L^(-α/2)(f)(x)-L^(-α/2)(bf)(x).We give applications to large classes of differential operators such as the Schr¨odinger operators and second-order elliptic operators of divergence form.展开更多
Let L=-div(A▽) be a second order divergence form elliptic operator with bounded measurable coefficients in R^(n).We establish weighted L^(p) norm inequalities for commutators generated by √L and Lipschitz functions,...Let L=-div(A▽) be a second order divergence form elliptic operator with bounded measurable coefficients in R^(n).We establish weighted L^(p) norm inequalities for commutators generated by √L and Lipschitz functions,where the range of p is different from(1,∞),and we isolate the right class of weights introduced by Auscher and Martell.In this work,we use good-λ inequality with two parameters through the weighted boundedness of Riesz transforms ▽L^(-1/2).Our result recovers,in some sense,a previous result of Hofmann.展开更多
For the commutators of multilinear Calderón-Zygmund singular integral operators with B MO functions, the weak type weighted norm inequalities with respect to A_(→P) weights are obtained.
Based on the theory of variable exponents and BMO norms, we prove the vector-valued inequalities for commutators of singular integrals on both homogeneous and inhomogeneous Herz spaces where the two main indices are v...Based on the theory of variable exponents and BMO norms, we prove the vector-valued inequalities for commutators of singular integrals on both homogeneous and inhomogeneous Herz spaces where the two main indices are variable exponents.Furthermore, we show that a wide class of commutators generated by BMO functions and sublinear operators satisfy vector-valued inequalities.展开更多
[b,T] denotes the commutator of generalized Calderon-Zygmund operators T with Lipschitz function b, where b∈Lip_β(R^n),(0 <β≤1) and T is aθ(t)-type Calderón-Zygmund operator. The commutator [b,T] generate...[b,T] denotes the commutator of generalized Calderon-Zygmund operators T with Lipschitz function b, where b∈Lip_β(R^n),(0 <β≤1) and T is aθ(t)-type Calderón-Zygmund operator. The commutator [b,T] generated by b and T is defined by[b,T]f(x)=b(x)Tf(x)-T(bf)(x)=∫k(x,y)(b(x)-b(y))f(y)dy.In this paper, the authors discuss the boundedness of the commutator [b, T] on weighted Hardy spaces and weighted Herz type Hardy spaces and prove that [b,T] is bounded from H^p(ω~p) to L^q(ω~q), and from HK_(q1)^(α,p)(ω_1,ω_2^(q1)) to K_(q2)^(α,p)(ω_1,ω_2^(q2)). The results extend and generalize the well-known ones in [7].展开更多
文摘In this paper,we consider the boundedness on Triebel-Lizorkin spaces for the d-dimensional Calder´on commutator defined by TΩ,af(x)=p.v.∫R_(d)Ω(x−y)/|x−y|^(d+1)(a(x)−a(y))f(y)dy,where Ω is homogeneous of degree zero,integrable on Sd−1 and has a vanishing moment of order one,and a is a function on Rd such that∇a∈L^(∞)(R^(d)).We prove that if 1<p,q<∞andΩ∈L(log L)^(2 q)(S^(d−1))with q=max{1/q,1/q′},then TΩ,a is bounded on Triebel-Lizorkin spaces˙F_(p)^(0)q(R^(d)).
文摘This paper is devoted to studying the behaviors of the fractional type Marcinkiewicz integralsμΩ,βand the commutatorsμΩ,βb generated byμΩ,βwith b b∈Lloc(Rn)on weighted Hardy spaces.Under the assumption of that the homogeneous kernelΩsatisfies certain regularities,the authors obtain the boundedness ofμΩ,βfrom the weighted Hardy spaces Hωpp(Rn)to the weighted Lebesgue spaces Lωqq(Rn)for n/(n+β)≤<p≤1 with 1/q=1/p-β/n,as well as the same(Hωpp,Lωqq)-boudedness ofμΩ,βb when b belongs to BMOωp,p(Rn),which is a non-trivial subspace of BMO(Rn).
基金supported by the NNSF of China(12271483,11961056)the NSF of Jiangxi Province(20192BAB201004)+1 种基金supported by the“Xin-Miao”Program of Zhejiang Province(2021R415027)the Innovation Fund of ZUST(2020yjskc06).
文摘In this paper,the authors consider theω-type Calder´on-Zygmund operator T_(ω)and the commutator[b,T_(ω)]generated by a symbol function b on the Lorentz space L^(p,r)(X)over the homogeneous space(X,d,μ).The boundedness and the compactness of the commutator[b,T_(ω)]on Lorentz space L^(p,r)(X)are founded for any p∈(1,∞)and r∈[1,∞).
基金Tang Lin and Yang Dachun are supported in part by the NNSF and the SEDF of China.
文摘Let p ∈(n/(n + l), 1]. The authors investigate the (Hbp(Rn), Lp(Rn)-type and (Hbp,∞(Rn), Lp,∞ (Rn)-type continuities for the maximal operators associated with the commutators of Bochner-mesz operators with BMO(Rn ) functions, where Hbp(Rn ) and Hpb∞(Rn) are, respectively, the variants of the standard Hardy spaces and the standard weak Hardy spaces.
基金Supported by the Anhui Polytechnic University Foundation for Recruiting Talent(2011YQQ004)Supported by the Provincial Natural Science Research Project of Anhui Colleges(KJ2011A032)+1 种基金Supported by the Young Teachers Program of Anhui Province(2006jql042)Supported by the Grant for Younth of Anhui Polytechnic University (2010YQ047)
文摘In this paper,the authors introduce the central bounded oscillation space CBMO q (R n),let [b,T,α ] be the commutator generated by fractional integral operators with variable kernels and CBMO function,we establish the boundedness of [b,T,α ] on homogeneous Morrey-Herz spaces.
基金supported by NSFC(11471033),NCET of China(NCET-11-0574)the Fundamental Research Funds for the Central Universities(FRF-BR-16-011A)
文摘Let L=-div(A▽) be a second order divergence form elliptic operator, where A is an accretive, n×n matrix with bounded measurable complex coefficients on R^n. Let L^α/2 (0 <α< 1) denotes the fractional differential operator associated with L and (-△)^α/2b ∈ L^n/α(R^n). In this article, we prove that the commutator[b, L^α/2] is bounded from the homogenous Sobolev space Lα%2 (R^n) to L^2(R^n).
文摘This paper establishes some strong type and weak type estimates for commutator [b,I1] on Herz-type spaces, where b E BMO(Rn) and I1 is a fractional integration with O < l < n.
基金supported by the “Basic Innovation” Program of Graduate Students of Guangzhou University(2018GDJC-D01)the second author is supported by the National Natural Science Foundation of China(11861062,11661075 and 11561065)the third author is supported by the the National Natural Science Foundation of China(11671414).
文摘Let T be an anisotropic Calderón-Zygmund operator andφ:R^n×[0,∞)→[0,∞)be an anisotropic Musielak-Orlicz function withφ(x,·)being an Orlicz function andφ(·,t)being a Muckenhoupt A∞(A)weight.In this paper,our goal is to study two boundedness theorems for commutators of anisotropic Calderon-Zygmund operators.Precisely,when b∈BMOw(R^n,A)(a proper subspace of anisotropic bounded mean oscillation space BMO(R^n,A)),the commutator[b,T]is bounded from anisotropic weighted Hardy space H^1ω(R^n,A)to weighted Lebesgue space L^1ω(R^n)and when b∈BMO(R^n)(bounded mean oscillation space),the commutator[b,T]is bounded on Musielak-Orlicz space L^φ(R^n),which are extensions of the isotropic setting.
基金Supported by the Natural Science Foundation of Shandong Province(Nos.ZR2018PA004 and ZR2016AB07)the National Natural Science Foundation of China(Nos.11571306 and 11671363)
文摘In this paper, the behavior for commutators of a class of bilinear singular integral operators associated with non-smooth kernels on the product of weighted Lebesgue spaces is considered. By some new maximal functions to control the commutators of bilinear singular integral operators and CMO(Rn) functions, compactness for the commutators is proved.
文摘In this paper, we study the boundedness of the fractional integral operator and their commutator on Herz spaecs with two variable exponents . By using the properties of the variable exponents Lebesgue spaces, the boundedness of the fractional integral operator and their commutator generated by Lipschitz function is obtained on those Herz spaces.
基金Supported by the National Natural Science Foundation of China(Grant No.11661075).
文摘In this paper, the authors establish the(L^p(μ), L^q(μ))-type estimate for fractional commutator generated by fractional integral operators Tα with Lipschitz functions(b ∈ Lipβ(μ)),where 1 < p < 1/(α + β) and 1/q = 1/p-(α + β), and obtain their weak(L^1(μ), L^(1/(1-α-β))(μ))-type. Moreover, the authors also consider the boundedness in the case that 1/(α+β) < p < 1/α,1/α≤ p ≤∞ and the endpoint cases, namely, p = 1/(α + β).
文摘Our aim in this paper is to prove the boundedness of commutators of Calderón-Zygmund operator with the Lipschitz function or BOM function on Herz-type Hardy space with variable exponent.
文摘The commutators of oscillatory singular integral operators with homogeneous kernel are studied, where Ω is homogeneous of degree zero, has mean value zero on the unit sphere. It is proved that Ω∈L (logL)(k+1)(S(n-1)) is a sufficient condition under which the k-th order commutator is bounded on L2(Rn).
基金supported by the NNSF of China(11571306)supported by the NNSF of China(11271330 and 11671363)supported by the NNSF of China(11371370)
文摘Let α∈0,(n-1)/2 and T~α be the Bochner-Riesz operator of order α. In this paper, for n = 2 and n ≥ 3, the compactness on Lebesgue spaces and Morrey spaces are considered for the commutator of Bochner-Riesz operator generated by CMO(R^n) function and T~α.
基金The Science and Technology Research(Z2014057)of Higher Education in Hebei Provincethe Doctoral Foundation(L2015B05)of Hebei Normal Universitythe NSF(A2015403040)of Hebei Province
文摘Let L be the infinitesimal generator of an analytic semigroup on L^2(R^n)with pointwise upper bounds on heat kernel,and denote by L^(-α/2)the fractional integrals of L.For a BMO function b(x),we show a weak type Llog L estimate of the commutators [b,L^(-α/2)](f)(x) = b(x)L^(-α/2)(f)(x)-L^(-α/2)(bf)(x).We give applications to large classes of differential operators such as the Schr¨odinger operators and second-order elliptic operators of divergence form.
基金supported by NSFC(1187109611471033)+4 种基金supported by NSFC(113710571147103311571160)SRFDP(20130003110003)the Fundamental Research Funds for the Central Universities(2014KJJCA10)。
文摘Let L=-div(A▽) be a second order divergence form elliptic operator with bounded measurable coefficients in R^(n).We establish weighted L^(p) norm inequalities for commutators generated by √L and Lipschitz functions,where the range of p is different from(1,∞),and we isolate the right class of weights introduced by Auscher and Martell.In this work,we use good-λ inequality with two parameters through the weighted boundedness of Riesz transforms ▽L^(-1/2).Our result recovers,in some sense,a previous result of Hofmann.
基金supported by the Natural Science Foundation of Hebei Province (A2014205069)
文摘For the commutators of multilinear Calderón-Zygmund singular integral operators with B MO functions, the weak type weighted norm inequalities with respect to A_(→P) weights are obtained.
文摘Based on the theory of variable exponents and BMO norms, we prove the vector-valued inequalities for commutators of singular integrals on both homogeneous and inhomogeneous Herz spaces where the two main indices are variable exponents.Furthermore, we show that a wide class of commutators generated by BMO functions and sublinear operators satisfy vector-valued inequalities.
文摘[b,T] denotes the commutator of generalized Calderon-Zygmund operators T with Lipschitz function b, where b∈Lip_β(R^n),(0 <β≤1) and T is aθ(t)-type Calderón-Zygmund operator. The commutator [b,T] generated by b and T is defined by[b,T]f(x)=b(x)Tf(x)-T(bf)(x)=∫k(x,y)(b(x)-b(y))f(y)dy.In this paper, the authors discuss the boundedness of the commutator [b, T] on weighted Hardy spaces and weighted Herz type Hardy spaces and prove that [b,T] is bounded from H^p(ω~p) to L^q(ω~q), and from HK_(q1)^(α,p)(ω_1,ω_2^(q1)) to K_(q2)^(α,p)(ω_1,ω_2^(q2)). The results extend and generalize the well-known ones in [7].