This paper is devoted to studying the commutators of the multilinear singular integral operators with the non-smooth kernels and the weighted Lipschitz functions. Some mapping properties for two types of commutators o...This paper is devoted to studying the commutators of the multilinear singular integral operators with the non-smooth kernels and the weighted Lipschitz functions. Some mapping properties for two types of commutators on the weighted Lebesgue spaces, which extend and generalize some previous results, are obtained.展开更多
In this paper, we establish two weighted integral inequalities for commutators of fractional Hardy operators with Besov-Lipschitz functions. The main result is that this kind of commutator, denoted by H^ab, is bounded...In this paper, we establish two weighted integral inequalities for commutators of fractional Hardy operators with Besov-Lipschitz functions. The main result is that this kind of commutator, denoted by H^ab, is bounded from L^Pxy （R＋） to L^qxδ （R＋） with the bound explicitly worked out.展开更多
In this paper, we establish the weighted <span style="white-space:nowrap;"><em>L</em><sup><em>p</em></sup></span> <span style="white-space:nowrap;"...In this paper, we establish the weighted <span style="white-space:nowrap;"><em>L</em><sup><em>p</em></sup></span> <span style="white-space:nowrap;">(1 < <em>p</em> < ∞)</span> boundedness of variation operator for the commutators generated by one-sided Calderón-Zygmund singular integrals with Lipschitz functions.展开更多
In this article, we obtain the LP-boundedness of commutators of Lipschitz functions and singular integrals with non-smooth kernels on Euclidean spaces.
Let 6=(bi,b2,...,bm)be a collection of locally integrable functions and T,the com-mutator of multilinear singular integral operator T.Denote by L(δ)and L(δ(·))the Lipschitz spaces and the variable Lipschitz spa...Let 6=(bi,b2,...,bm)be a collection of locally integrable functions and T,the com-mutator of multilinear singular integral operator T.Denote by L(δ)and L(δ(·))the Lipschitz spaces and the variable Lipschitz spaces,respectively.The main purpose of this paper is to establish some new characterizations of the(variable)Lipschitz spaces in terms of the boundedness of multilinear commutator T∑b in the context of the variable exponent Lebesgue spaces,that is,the authors give the necessary and sufficient conditions for bj(j=1,2,...,m)to be L(δ)or L(δ(·))via the boundedness of multilinear commutator from products of variable exponent Lebesgue spaces to variable exponent Lebesgue spaces.The authors do so by applying the Fourier series technique and some pointwise esti-mate for the commutators.The key tool in obtaining such pointwise estimate is a certain generalization of the classical sharp maximal operator.展开更多
Under the assumption that the underlying measure is a non-negative Radon measure which only satisfies some growth condition, the authors prove that for a class of commutators with Lipschitz functions which include com...Under the assumption that the underlying measure is a non-negative Radon measure which only satisfies some growth condition, the authors prove that for a class of commutators with Lipschitz functions which include commutators generated by Calderon-Zygrnund operators and Lipschitz functions as examples, their boundedness in Lebesgue spaces or the Hardy space H^1 （μ） is equivalent to some endpoint estimates satisfied by them. This result is new even when the underlying measure μ is the d-dimensional Lebesgue measure.展开更多
We discuss the relationship between Lipschitz functions and convex functions. By these relations, we give a sufficient condition for the set of points where Lipschitz functions on a Hilbert space is Frechet differenti...We discuss the relationship between Lipschitz functions and convex functions. By these relations, we give a sufficient condition for the set of points where Lipschitz functions on a Hilbert space is Frechet differentiate to be residual.展开更多
In this paper we provide a complete description of linear biseparating maps between spaces lip0（X^a, E） of Banach-valued little Lipschitz functions vanishing at infinity on locally com-pact HSlder metric spaces X^a=...In this paper we provide a complete description of linear biseparating maps between spaces lip0（X^a, E） of Banach-valued little Lipschitz functions vanishing at infinity on locally com-pact HSlder metric spaces X^a=（X,dx^a） with 0〈a〈1.Namely, it is proved that any linear bijection T ： lip0（X^a,E）→lip0（Y^a,F）satisfying that ｜｜Tf（y）｜｜F｜｜Tg（y）｜｜F= 0 for all y ∈ Y if and only if ｜｜f（x）｜｜E｜｜g（x）｜｜E=0 for all x E X, is a weighted composition operator of the form Tf（y） = h（y）（f（φ（y）））, where φ is a homeomorphism from Y onto X and h is a map from Y into the set of all linear bijections from E onto F. Moreover, T is continuous if and only if h（y） is continuous for all y ∈ Y. In this case, φ becomes a locally Lipschitz homeomorphism and h a locally Lipschitz map from Y^a into the space of all continuous linear bijections from E onto F with the metric induced by the operator canonical norm. This enables us to study the automatic continuity of T and the existence of discontinuous linear biseparating maps.展开更多
The differentiability of a norm of a Banach space may be characterized by its unit sphere.This paper generalizes these geometric conditions of norm’s differentiability to the case of a regular locally Lipschitz funct...The differentiability of a norm of a Banach space may be characterized by its unit sphere.This paper generalizes these geometric conditions of norm’s differentiability to the case of a regular locally Lipschitz function.展开更多
In this paper,we study some basic properties on Lipschitz star bodies,such as the equivalence between Lipschitz star bodies and star bodies with respect to a ball,the equivalence between the convergence of Lipschitz s...In this paper,we study some basic properties on Lipschitz star bodies,such as the equivalence between Lipschitz star bodies and star bodies with respect to a ball,the equivalence between the convergence of Lipschitz star bodies with respect to Hausdorff distance and the convergence of Lipschtz star bodies with respect to radial distance,and the convergence of Steiner symmetrizations of Lipschitz star bodies.展开更多
SINCE Namioka and Phelps, starting with Asplund’s pioneering work, introduced the no-tion of Asplund spaces (those are Banach spaces on which every continuous convex function isFrechet differentiable on a dense G_δ ...SINCE Namioka and Phelps, starting with Asplund’s pioneering work, introduced the no-tion of Asplund spaces (those are Banach spaces on which every continuous convex function isFrechet differentiable on a dense G_δ subset) and proved that the dual of an Asplund space hasthe Radon-Nikodym property (RNP), the study of differentiability properties of functions oninfinite dimensional spaces has continued widely and deeply (see, for instance, Phelps andGiles). The research attained a great achievment after Stegall’s theorem: If the dualspace E~* has the RNP, then E is an Asplund space. Because of the N-Ph-S theorem, we展开更多
When all the involved data in indefinite quadratic programs change simultaneously, we show the locally Lipschtiz continuity of the KKT set of the quadratic programming problem firstly, then we establish the locally Li...When all the involved data in indefinite quadratic programs change simultaneously, we show the locally Lipschtiz continuity of the KKT set of the quadratic programming problem firstly, then we establish the locally Lipschtiz continuity of the KKT solution set. Finally, the similar conclusion for the corresponding optimal value function is obtained.展开更多
Let G be an Abelian group and letρ：G×G→[0,∞） be a metric on G. Let E be a normed space. We prove that under some conditions if f：G→E is an odd function and Cx：G→E defined by Cx（y）：=2 f （x＋y）＋2 f ...Let G be an Abelian group and letρ：G×G→[0,∞） be a metric on G. Let E be a normed space. We prove that under some conditions if f：G→E is an odd function and Cx：G→E defined by Cx（y）：=2 f （x＋y）＋2 f （x-y）＋12 f （x）-f （2x＋y）-f （2x-y） is a cubic function for all x∈G, then there exists a cubic function C：G→E such that f？C is Lipschitz. Moreover, we investigate the stability of cubic functional equation 2 f （x＋y）＋2 f （x-y）＋12 f （x）-f （2x＋y）-f （2x-y）=0 on Lipschitz spaces.展开更多
Let X (t)(t∈R^N) be a d-dimensional fractional Brownian motion. A contiunous function f:R^N→R^d is called a polar function of X(t)(t∈R^N) if P{ t∈R^N\{0},X(t)=t(t)}=0. In this paper, the characteristies of the cla...Let X (t)(t∈R^N) be a d-dimensional fractional Brownian motion. A contiunous function f:R^N→R^d is called a polar function of X(t)(t∈R^N) if P{ t∈R^N\{0},X(t)=t(t)}=0. In this paper, the characteristies of the class of polar functions are studied. Our theorem 1 improves the previous results of Graversen and Legall. Theorem2 solves a problem of Legall (1987) on Brownian motion.展开更多
基金Supported by the National Natural Science Foundation of China (10771054,11071200)the NFS of Fujian Province of China (No. 2010J01013)
文摘This paper is devoted to studying the commutators of the multilinear singular integral operators with the non-smooth kernels and the weighted Lipschitz functions. Some mapping properties for two types of commutators on the weighted Lebesgue spaces, which extend and generalize some previous results, are obtained.
基金Supported in part by the Natural Science Foundation of China under the Grant 10771221Natural Science Foundation of Beijing under the Grant 1092004
文摘In this paper, we establish two weighted integral inequalities for commutators of fractional Hardy operators with Besov-Lipschitz functions. The main result is that this kind of commutator, denoted by H^ab, is bounded from L^Pxy （R＋） to L^qxδ （R＋） with the bound explicitly worked out.
文摘In this paper, we establish the weighted <span style="white-space:nowrap;"><em>L</em><sup><em>p</em></sup></span> <span style="white-space:nowrap;">(1 < <em>p</em> < ∞)</span> boundedness of variation operator for the commutators generated by one-sided Calderón-Zygmund singular integrals with Lipschitz functions.
文摘In this article, we obtain the LP-boundedness of commutators of Lipschitz functions and singular integrals with non-smooth kernels on Euclidean spaces.
基金Supported by the National Natural Science Foundation of China(Grant No.11571160)the Research Funds for the Educational Committee of Heilongjiang(Grant No.2019-KYYWF-0909)the Reform and Development Foundation for Local Colleges and Universities of the Central Government(Grant No.2020YQ07)。
文摘Let 6=(bi,b2,...,bm)be a collection of locally integrable functions and T,the com-mutator of multilinear singular integral operator T.Denote by L(δ)and L(δ(·))the Lipschitz spaces and the variable Lipschitz spaces,respectively.The main purpose of this paper is to establish some new characterizations of the(variable)Lipschitz spaces in terms of the boundedness of multilinear commutator T∑b in the context of the variable exponent Lebesgue spaces,that is,the authors give the necessary and sufficient conditions for bj(j=1,2,...,m)to be L(δ)or L(δ(·))via the boundedness of multilinear commutator from products of variable exponent Lebesgue spaces to variable exponent Lebesgue spaces.The authors do so by applying the Fourier series technique and some pointwise esti-mate for the commutators.The key tool in obtaining such pointwise estimate is a certain generalization of the classical sharp maximal operator.
基金Project supported by the National Natural Science Foundation of China (No. 10271015)the Program for New Century Excellent Talents in Universities of China (No. NCET-04-0142).
文摘Under the assumption that the underlying measure is a non-negative Radon measure which only satisfies some growth condition, the authors prove that for a class of commutators with Lipschitz functions which include commutators generated by Calderon-Zygrnund operators and Lipschitz functions as examples, their boundedness in Lebesgue spaces or the Hardy space H^1 （μ） is equivalent to some endpoint estimates satisfied by them. This result is new even when the underlying measure μ is the d-dimensional Lebesgue measure.
文摘We discuss the relationship between Lipschitz functions and convex functions. By these relations, we give a sufficient condition for the set of points where Lipschitz functions on a Hilbert space is Frechet differentiate to be residual.
基金supported by Junta de Andalucia grants FQM-1438 and FQM-3737
文摘In this paper we provide a complete description of linear biseparating maps between spaces lip0（X^a, E） of Banach-valued little Lipschitz functions vanishing at infinity on locally com-pact HSlder metric spaces X^a=（X,dx^a） with 0〈a〈1.Namely, it is proved that any linear bijection T ： lip0（X^a,E）→lip0（Y^a,F）satisfying that ｜｜Tf（y）｜｜F｜｜Tg（y）｜｜F= 0 for all y ∈ Y if and only if ｜｜f（x）｜｜E｜｜g（x）｜｜E=0 for all x E X, is a weighted composition operator of the form Tf（y） = h（y）（f（φ（y）））, where φ is a homeomorphism from Y onto X and h is a map from Y into the set of all linear bijections from E onto F. Moreover, T is continuous if and only if h（y） is continuous for all y ∈ Y. In this case, φ becomes a locally Lipschitz homeomorphism and h a locally Lipschitz map from Y^a into the space of all continuous linear bijections from E onto F with the metric induced by the operator canonical norm. This enables us to study the automatic continuity of T and the existence of discontinuous linear biseparating maps.
基金Supported by the National Natural Science Foundation of China
文摘The differentiability of a norm of a Banach space may be characterized by its unit sphere.This paper generalizes these geometric conditions of norm’s differentiability to the case of a regular locally Lipschitz function.
基金supported by the NSFC(11971080,KJQN202000838)the funds of the Basic and Advanced Research Project of CQ CSTC(cstc2018jcyj AX0790,cstc2020jcyj-msxm X0328)+1 种基金supported by Project funded by the China Postdoctoral Science Foundation(2019TQ0097)the Science and Technology Commission of Shanghai Municipality(22DZ2229014)。
文摘In this paper,we study some basic properties on Lipschitz star bodies,such as the equivalence between Lipschitz star bodies and star bodies with respect to a ball,the equivalence between the convergence of Lipschitz star bodies with respect to Hausdorff distance and the convergence of Lipschtz star bodies with respect to radial distance,and the convergence of Steiner symmetrizations of Lipschitz star bodies.
文摘SINCE Namioka and Phelps, starting with Asplund’s pioneering work, introduced the no-tion of Asplund spaces (those are Banach spaces on which every continuous convex function isFrechet differentiable on a dense G_δ subset) and proved that the dual of an Asplund space hasthe Radon-Nikodym property (RNP), the study of differentiability properties of functions oninfinite dimensional spaces has continued widely and deeply (see, for instance, Phelps andGiles). The research attained a great achievment after Stegall’s theorem: If the dualspace E~* has the RNP, then E is an Asplund space. Because of the N-Ph-S theorem, we
基金Supported by the National Natural Science Foundation of China(10571141,70971109,71371152)supported by the Talents Fund of Xi’an Polytechnic University(BS1320)the Mathematics Discipline Development Fund of Xi’an Ploytechnic University(107090701)
文摘When all the involved data in indefinite quadratic programs change simultaneously, we show the locally Lipschtiz continuity of the KKT set of the quadratic programming problem firstly, then we establish the locally Lipschtiz continuity of the KKT solution set. Finally, the similar conclusion for the corresponding optimal value function is obtained.
文摘Let G be an Abelian group and letρ：G×G→[0,∞） be a metric on G. Let E be a normed space. We prove that under some conditions if f：G→E is an odd function and Cx：G→E defined by Cx（y）：=2 f （x＋y）＋2 f （x-y）＋12 f （x）-f （2x＋y）-f （2x-y） is a cubic function for all x∈G, then there exists a cubic function C：G→E such that f？C is Lipschitz. Moreover, we investigate the stability of cubic functional equation 2 f （x＋y）＋2 f （x-y）＋12 f （x）-f （2x＋y）-f （2x-y）=0 on Lipschitz spaces.
文摘Let X (t)(t∈R^N) be a d-dimensional fractional Brownian motion. A contiunous function f:R^N→R^d is called a polar function of X(t)(t∈R^N) if P{ t∈R^N\{0},X(t)=t(t)}=0. In this paper, the characteristies of the class of polar functions are studied. Our theorem 1 improves the previous results of Graversen and Legall. Theorem2 solves a problem of Legall (1987) on Brownian motion.
