In this paper, we consider minimal L^(2) integrals on the sublevel sets of plurisubharmonic functions on weakly pseudoconvex K?hler manifolds with Lebesgue measurable gain related to modules at boundary points of the ...In this paper, we consider minimal L^(2) integrals on the sublevel sets of plurisubharmonic functions on weakly pseudoconvex K?hler manifolds with Lebesgue measurable gain related to modules at boundary points of the sublevel sets, and establish a concavity property of the minimal L^(2) integrals. As applications, we present a necessary condition for the concavity degenerating to linearity, a concavity property related to modules at inner points of the sublevel sets, an optimal support function related to modules, a strong openness property of modules and a twisted version, and an effectiveness result of the strong openness property of modules.展开更多
In this article,we consider a modified version of minimal L^(2) integrals on sublevel sets of plurisubharmonic functions related to modules at boundary points,and obtain a concavity property of the modified version.As...In this article,we consider a modified version of minimal L^(2) integrals on sublevel sets of plurisubharmonic functions related to modules at boundary points,and obtain a concavity property of the modified version.As applications,we give characterizations for the concavity degenerating to linearity on open Riemann surfaces and on fibrations over open Riemann surfaces.展开更多
In this article,we investigate the(big) Hankel operator H_(f) on the Hardy spaces of bounded strongly pseudoconvex domains Ω in C^(n).We observe that H_(f ) is bounded on H~p(Ω)(1 <p <∞) if f belongs to BMO a...In this article,we investigate the(big) Hankel operator H_(f) on the Hardy spaces of bounded strongly pseudoconvex domains Ω in C^(n).We observe that H_(f ) is bounded on H~p(Ω)(1 <p <∞) if f belongs to BMO and we obtain some characterizations for Hf on H^(2)(Ω) of other pseudoconvex domains.In these arguments,Amar's L^(p)-estimations and Berndtsson's L^(2)-estimations for solutions of the ■_(b)-equation play a crucial role.In addition,we solve Gleason's problem for Hardy spaces H^(p)(Ω)(1 ≤p≤∞) of bounded strongly pseudoconvex domains.展开更多
This paper studies an online distributed optimization problem over multi-agent systems.In this problem,the goal of agents is to cooperatively minimize the sum of locally dynamic cost functions.Different from most exis...This paper studies an online distributed optimization problem over multi-agent systems.In this problem,the goal of agents is to cooperatively minimize the sum of locally dynamic cost functions.Different from most existing works on distributed optimization,here we consider the case where the cost function is strongly pseudoconvex and real gradients of objective functions are not available.To handle this problem,an online zeroth-order stochastic optimization algorithm involving the single-point gradient estimator is proposed.Under the algorithm,each agent only has access to the information associated with its own cost function and the estimate of the gradient,and exchange local state information with its immediate neighbors via a time-varying digraph.The performance of the algorithm is measured by the expectation of dynamic regret.Under mild assumptions on graphs,we prove that if the cumulative deviation of minimizer sequence grows within a certain rate,then the expectation of dynamic regret grows sublinearly.Finally,a simulation example is given to illustrate the validity of our results.展开更多
This paper focuses on the online distributed optimization problem based on multi-agent systems. In this problem, each agent can only access its own cost function and a convex set, and can only exchange local state inf...This paper focuses on the online distributed optimization problem based on multi-agent systems. In this problem, each agent can only access its own cost function and a convex set, and can only exchange local state information with its current neighbors through a time-varying digraph. In addition, the agents do not have access to the information about the current cost functions until decisions are made. Different from most existing works on online distributed optimization, here we consider the case where the cost functions are strongly pseudoconvex and real gradients of the cost functions are not available. To handle this problem, a random gradient-free online distributed algorithm involving the multi-point gradient estimator is proposed. Of particular interest is that under the proposed algorithm, each agent only uses the estimation information of gradients instead of the real gradient information to make decisions. The dynamic regret is employed to measure the proposed algorithm. We prove that if the cumulative deviation of the minimizer sequence grows within a certain rate, then the expectation of dynamic regret increases sublinearly. Finally, a simulation example is given to corroborate the validity of our results.展开更多
In this paper, we study the integral solution operators for the -equations on pseudoconvex domains. As a generalization of [1] for the -dequations on pseudoconvex domains with boundary of class C∞, we obtain the ex...In this paper, we study the integral solution operators for the -equations on pseudoconvex domains. As a generalization of [1] for the -dequations on pseudoconvex domains with boundary of class C∞, we obtain the explicit integral operator solutions of C -form for the -equations on pseudoconvex open sets with boundary of Ck (k≥0) and the sup-norm estimates of which solutions have similar as that [1] in form.展开更多
A new Koppelman-Leray-Norguet formula of (p,q) differential forms for a strictly pseudoconvex polyhedron with not necessarily smooth boundary on a Stein manifold is obtained, and an integral representation for the sol...A new Koppelman-Leray-Norguet formula of (p,q) differential forms for a strictly pseudoconvex polyhedron with not necessarily smooth boundary on a Stein manifold is obtained, and an integral representation for the solution of -equation on this domain which does not involve integrals on boundary is given, so one can avoid complex estimates of boundary integrals.展开更多
The concepts of alpha-order Clarke's derivative, alpha-order Adjacent derivative and alpha-order G.Bouligand derivative of set-valued mappings are introduced, their properties are studied, with which the Fritz Joh...The concepts of alpha-order Clarke's derivative, alpha-order Adjacent derivative and alpha-order G.Bouligand derivative of set-valued mappings are introduced, their properties are studied, with which the Fritz John optimality condition of set-valued vector optimization is established. Finally, under the assumption of pseudoconvexity, the optimality condition is proved to be sufficient.展开更多
Kytmanov and Myslivets gave a special Cauchy principal value of the singular integral on the bounded strictly pseudoconvex domain with smooth boundary. By means of this Cauchy integral principal value, the correspondi...Kytmanov and Myslivets gave a special Cauchy principal value of the singular integral on the bounded strictly pseudoconvex domain with smooth boundary. By means of this Cauchy integral principal value, the corresponding singular integral and a composition formula are obtained. This composition formula is quite different from usual ones in form. As an application, the corresponding singular integral equation and the system of singular integral equations are discussed as well.展开更多
In this paper, we discuss some recent studies on the complex structure of an isolated normal singularity by using the information from its link. We also give some open problems to be further pursued.
A necessary and sufficient condition for the boundedness of the operator: $(T_{s,u,u} f)(\xi ) = h^{u + \tfrac{v}{a}} (\xi )\smallint _{\Omega _a } h^s (\xi ')K_{s,u,v} (\xi ,\xi ')f(\xi ')dv(\xi ') on...A necessary and sufficient condition for the boundedness of the operator: $(T_{s,u,u} f)(\xi ) = h^{u + \tfrac{v}{a}} (\xi )\smallint _{\Omega _a } h^s (\xi ')K_{s,u,v} (\xi ,\xi ')f(\xi ')dv(\xi ') on L^p (\Omega _a ,dv_\lambda ),1< p< \infty $ , is obtained, where $\Omega _a = \left\{ {\xi = (z,w) \in \mathbb{C}^{n + m} :z \in \mathbb{C}^n ,w \in \mathbb{C}^m ,|z|^2 + |w|^{2/a}< 1} \right\},h(\xi ) = (1 - |z|^2 )^a - |w|^2 $ andK x,u,v (ξ,ξ′).This generalizes the works in literature from the unit ball or unit disc to the weakly pseudoconvex domain ω a . As an appli cation, it is proved thatf?L H p (ω a ,dv λ) implies $h\tfrac{{|a|}}{a} + |\beta |(\xi )D_2^a D_z^\beta f \in L^p (\Omega _a ,dv_\lambda ),1 \leqslant p< \infty $ , for any multi-indexa=(α1,?,α n and ? = (?1, —?). An interesting question is whether the converse holds.展开更多
The purpose of this paper is threefold.(i) To explain the effective Kohn algorithm for multipliers in the complex Neumann problem and its difference with the full-real-radical Kohn algorithm, especially in the context...The purpose of this paper is threefold.(i) To explain the effective Kohn algorithm for multipliers in the complex Neumann problem and its difference with the full-real-radical Kohn algorithm, especially in the context of an example of Catlin-D'Angelo concerning the ineffectiveness of the latter.(ii) To extend the techniques of multiplier ideal sheaves for the complex Neumann problem to general systems of partial differential equations.(iii) To present a new procedure of generation of multipliers in the complex Neumann problem as a special case of the multiplier ideal sheaves techniques for general systems of partial differential equations.展开更多
In this paper we obtain non-isotropic weighted Lp estimates with the boundary distance weight function for the-equation on piecewise smooth strictly pseudoconvex domains under a hypoth- esis of complex transversality ...In this paper we obtain non-isotropic weighted Lp estimates with the boundary distance weight function for the-equation on piecewise smooth strictly pseudoconvex domains under a hypoth- esis of complex transversality in Cn using the explicit formula of solutions by Berndtsson-Andersson.展开更多
We discuss some recent results on interpolation problems for weighted Hrmander’s algebras of holomorphic functions in several complex variables, and also give a sharp estimate on counting functions of interpolating v...We discuss some recent results on interpolation problems for weighted Hrmander’s algebras of holomorphic functions in several complex variables, and also give a sharp estimate on counting functions of interpolating varieties.展开更多
The purpose of this paper is to complement the results by Lanzani and Stein （2017） by showing thedense definability of the Cauchy-Leray transform for the domains that give the counter-examples of Lanzani andStein （...The purpose of this paper is to complement the results by Lanzani and Stein （2017） by showing thedense definability of the Cauchy-Leray transform for the domains that give the counter-examples of Lanzani andStein （2017）, where LP-boundedness is shown to fail when either the ＂near＂ C2 boundary regularity, or the strongC-linear convexity assumption is dropped.展开更多
The purpose of this paper is to introduce second order (K, F)-pseudoconvex and second order strongly (K, F)- pseudoconvex functions which are a generalization of cone-pseudoconvex and strongly cone-pseudoconvex functi...The purpose of this paper is to introduce second order (K, F)-pseudoconvex and second order strongly (K, F)- pseudoconvex functions which are a generalization of cone-pseudoconvex and strongly cone-pseudoconvex functions. A pair of second order symmetric dual multiobjective nonlinear programs is formulated by using the considered functions. Furthermore, the weak, strong and converse duality theorems for this pair are established. Finally, a self duality theorem is given.展开更多
The theory of iterates of holomorphic maps is a very active topic in recent years. The well-known Denjoy-Wolff theorem characterized the asymptotic behavior of the iterate sequences of holomorphic self-maps in the uni...The theory of iterates of holomorphic maps is a very active topic in recent years. The well-known Denjoy-Wolff theorem characterized the asymptotic behavior of the iterate sequences of holomorphic self-maps in the unit disc △C. Subsequent authors extended this theorem to some special domains of C^n(n】1). Briefly speaking, the research展开更多
In this paper,we present the concavity of the minimal L^(2)integrals related to multiplier ideal sheaves on the weakly pseudoconvex Kahler manifolds,which implies the sharp effectiveness results of the strong openness...In this paper,we present the concavity of the minimal L^(2)integrals related to multiplier ideal sheaves on the weakly pseudoconvex Kahler manifolds,which implies the sharp effectiveness results of the strong openness conjecture and a conjecture posed by Demailly and Kollar(2001)on weakly pseudoconvex Kahler manifolds.We obtain the relation between the concavity and the L^(2)extension theorem.展开更多
基金supported by National Key R&D Program of China (Grant No. 2021YFA1003100)supported by National Natural Science Foundation of China (Grant Nos. 11825101, 11522101, and 11431013)+1 种基金supported by the Talent Fund of Beijing Jiaotong Universitysupported by China Postdoctoral Science Foundation (Grant Nos. BX20230402 and 2023M743719)。
文摘In this paper, we consider minimal L^(2) integrals on the sublevel sets of plurisubharmonic functions on weakly pseudoconvex K?hler manifolds with Lebesgue measurable gain related to modules at boundary points of the sublevel sets, and establish a concavity property of the minimal L^(2) integrals. As applications, we present a necessary condition for the concavity degenerating to linearity, a concavity property related to modules at inner points of the sublevel sets, an optimal support function related to modules, a strong openness property of modules and a twisted version, and an effectiveness result of the strong openness property of modules.
基金supported by National Key R&D Program of China(Grant No.2021YFA1003100)supported by NSFC(Grant Nos.11825101,11522101 and 11431013)+1 种基金supported by the Talent Fund of Beijing Jiaotong Universitysupported by China Postdoctoral Science Foundation(Grant Nos.BX20230402 and 2023M743719)。
文摘In this article,we consider a modified version of minimal L^(2) integrals on sublevel sets of plurisubharmonic functions related to modules at boundary points,and obtain a concavity property of the modified version.As applications,we give characterizations for the concavity degenerating to linearity on open Riemann surfaces and on fibrations over open Riemann surfaces.
基金supported by the National Natural Science Foundation of China(12271101)。
文摘In this article,we investigate the(big) Hankel operator H_(f) on the Hardy spaces of bounded strongly pseudoconvex domains Ω in C^(n).We observe that H_(f ) is bounded on H~p(Ω)(1 <p <∞) if f belongs to BMO and we obtain some characterizations for Hf on H^(2)(Ω) of other pseudoconvex domains.In these arguments,Amar's L^(p)-estimations and Berndtsson's L^(2)-estimations for solutions of the ■_(b)-equation play a crucial role.In addition,we solve Gleason's problem for Hardy spaces H^(p)(Ω)(1 ≤p≤∞) of bounded strongly pseudoconvex domains.
基金Supported by National Natural Science Foundation of China(62103169,51875380)China Postdoctoral Science Foundation(2021M691313)。
文摘This paper studies an online distributed optimization problem over multi-agent systems.In this problem,the goal of agents is to cooperatively minimize the sum of locally dynamic cost functions.Different from most existing works on distributed optimization,here we consider the case where the cost function is strongly pseudoconvex and real gradients of objective functions are not available.To handle this problem,an online zeroth-order stochastic optimization algorithm involving the single-point gradient estimator is proposed.Under the algorithm,each agent only has access to the information associated with its own cost function and the estimate of the gradient,and exchange local state information with its immediate neighbors via a time-varying digraph.The performance of the algorithm is measured by the expectation of dynamic regret.Under mild assumptions on graphs,we prove that if the cumulative deviation of minimizer sequence grows within a certain rate,then the expectation of dynamic regret grows sublinearly.Finally,a simulation example is given to illustrate the validity of our results.
基金supported by the National Natural Science Foundation of China(Nos.62103169,51875380)the China Postdoctoral Science Foundation(No.2021M691313).
文摘This paper focuses on the online distributed optimization problem based on multi-agent systems. In this problem, each agent can only access its own cost function and a convex set, and can only exchange local state information with its current neighbors through a time-varying digraph. In addition, the agents do not have access to the information about the current cost functions until decisions are made. Different from most existing works on online distributed optimization, here we consider the case where the cost functions are strongly pseudoconvex and real gradients of the cost functions are not available. To handle this problem, a random gradient-free online distributed algorithm involving the multi-point gradient estimator is proposed. Of particular interest is that under the proposed algorithm, each agent only uses the estimation information of gradients instead of the real gradient information to make decisions. The dynamic regret is employed to measure the proposed algorithm. We prove that if the cumulative deviation of the minimizer sequence grows within a certain rate, then the expectation of dynamic regret increases sublinearly. Finally, a simulation example is given to corroborate the validity of our results.
文摘In this paper, we study the integral solution operators for the -equations on pseudoconvex domains. As a generalization of [1] for the -dequations on pseudoconvex domains with boundary of class C∞, we obtain the explicit integral operator solutions of C -form for the -equations on pseudoconvex open sets with boundary of Ck (k≥0) and the sup-norm estimates of which solutions have similar as that [1] in form.
基金Supported by the National Natural Science Foundation and Mathematical "Tian Yuan" Foundation of China and the Natural Science Foundation of Fujian (Grant No. 10271097, TY10126033, F0110012)
文摘A new Koppelman-Leray-Norguet formula of (p,q) differential forms for a strictly pseudoconvex polyhedron with not necessarily smooth boundary on a Stein manifold is obtained, and an integral representation for the solution of -equation on this domain which does not involve integrals on boundary is given, so one can avoid complex estimates of boundary integrals.
基金the National Natural Science Foundation(69972036) and the Natural Science Foundation of Shanxi province(995L02)
文摘The concepts of alpha-order Clarke's derivative, alpha-order Adjacent derivative and alpha-order G.Bouligand derivative of set-valued mappings are introduced, their properties are studied, with which the Fritz John optimality condition of set-valued vector optimization is established. Finally, under the assumption of pseudoconvexity, the optimality condition is proved to be sufficient.
基金the Natural Science Foundation of Zhejiang Province (Y605149)the National Natural Science Foundation of China (10571156)
文摘Kytmanov and Myslivets gave a special Cauchy principal value of the singular integral on the bounded strictly pseudoconvex domain with smooth boundary. By means of this Cauchy integral principal value, the corresponding singular integral and a composition formula are obtained. This composition formula is quite different from usual ones in form. As an application, the corresponding singular integral equation and the system of singular integral equations are discussed as well.
文摘In this paper, we discuss some recent studies on the complex structure of an isolated normal singularity by using the information from its link. We also give some open problems to be further pursued.
文摘A necessary and sufficient condition for the boundedness of the operator: $(T_{s,u,u} f)(\xi ) = h^{u + \tfrac{v}{a}} (\xi )\smallint _{\Omega _a } h^s (\xi ')K_{s,u,v} (\xi ,\xi ')f(\xi ')dv(\xi ') on L^p (\Omega _a ,dv_\lambda ),1< p< \infty $ , is obtained, where $\Omega _a = \left\{ {\xi = (z,w) \in \mathbb{C}^{n + m} :z \in \mathbb{C}^n ,w \in \mathbb{C}^m ,|z|^2 + |w|^{2/a}< 1} \right\},h(\xi ) = (1 - |z|^2 )^a - |w|^2 $ andK x,u,v (ξ,ξ′).This generalizes the works in literature from the unit ball or unit disc to the weakly pseudoconvex domain ω a . As an appli cation, it is proved thatf?L H p (ω a ,dv λ) implies $h\tfrac{{|a|}}{a} + |\beta |(\xi )D_2^a D_z^\beta f \in L^p (\Omega _a ,dv_\lambda ),1 \leqslant p< \infty $ , for any multi-indexa=(α1,?,α n and ? = (?1, —?). An interesting question is whether the converse holds.
文摘The purpose of this paper is threefold.(i) To explain the effective Kohn algorithm for multipliers in the complex Neumann problem and its difference with the full-real-radical Kohn algorithm, especially in the context of an example of Catlin-D'Angelo concerning the ineffectiveness of the latter.(ii) To extend the techniques of multiplier ideal sheaves for the complex Neumann problem to general systems of partial differential equations.(iii) To present a new procedure of generation of multipliers in the complex Neumann problem as a special case of the multiplier ideal sheaves techniques for general systems of partial differential equations.
基金supported by the Korea Research Foundation Grant funded by Korea Government(MOEHRD,Basic Research Promotion Fund)(Grant No.KRF-2005-070-C00007)
文摘In this paper we obtain non-isotropic weighted Lp estimates with the boundary distance weight function for the-equation on piecewise smooth strictly pseudoconvex domains under a hypoth- esis of complex transversality in Cn using the explicit formula of solutions by Berndtsson-Andersson.
文摘We discuss some recent results on interpolation problems for weighted Hrmander’s algebras of holomorphic functions in several complex variables, and also give a sharp estimate on counting functions of interpolating varieties.
基金supported by the National Science Foundation of USA (Grant Nos. DMS1503612 (Lanzani) and DMS-1265524 (Stein))
文摘The purpose of this paper is to complement the results by Lanzani and Stein （2017） by showing thedense definability of the Cauchy-Leray transform for the domains that give the counter-examples of Lanzani andStein （2017）, where LP-boundedness is shown to fail when either the ＂near＂ C2 boundary regularity, or the strongC-linear convexity assumption is dropped.
文摘The purpose of this paper is to introduce second order (K, F)-pseudoconvex and second order strongly (K, F)- pseudoconvex functions which are a generalization of cone-pseudoconvex and strongly cone-pseudoconvex functions. A pair of second order symmetric dual multiobjective nonlinear programs is formulated by using the considered functions. Furthermore, the weak, strong and converse duality theorems for this pair are established. Finally, a self duality theorem is given.
文摘The theory of iterates of holomorphic maps is a very active topic in recent years. The well-known Denjoy-Wolff theorem characterized the asymptotic behavior of the iterate sequences of holomorphic self-maps in the unit disc △C. Subsequent authors extended this theorem to some special domains of C^n(n】1). Briefly speaking, the research
基金supported by National Natural Science Foundation of China (Grant Nos. 11825101, 11522101 and 11431013)
文摘In this paper,we present the concavity of the minimal L^(2)integrals related to multiplier ideal sheaves on the weakly pseudoconvex Kahler manifolds,which implies the sharp effectiveness results of the strong openness conjecture and a conjecture posed by Demailly and Kollar(2001)on weakly pseudoconvex Kahler manifolds.We obtain the relation between the concavity and the L^(2)extension theorem.
