Let L be a linear operator in L^2（R^n） and generate an analytic semigroup {e^-tL}t≥0 with kernel satisfying an upper bound estimate of Poisson type, whose decay is measured by θ（L） ∈ （0, ∞）. Let 4） be a pos...Let L be a linear operator in L^2（R^n） and generate an analytic semigroup {e^-tL}t≥0 with kernel satisfying an upper bound estimate of Poisson type, whose decay is measured by θ（L） ∈ （0, ∞）. Let 4） be a positive, continuous and strictly increasing function on （0, ∞）, which is of strictly critical lower type pФ （n/（n ＋ θ（L））, 1]. Denote by HФ, L（R^n） the Orlicz-Hardy space introduced in Jiang, Yang and Zhou＇s paper in 2009. If Ф is additionally of upper type 1 and subadditive, the authors then show that the Littlewood-Paley g-function gL maps HФ, L（R^n） continuously into LФ（R^n） and, moreover, the authors characterize HФ, L（R^n） in terms of the Littlewood-Paley gλ^＊-function with λ ∈ （n（2/pФ ＋ 1）, ∞）. If Ф is further slightly strengthened to be concave, the authors show that a generalized Riesz transform associated with L is bounded from HФ, L（R^n） to the Orlicz space L^Ф（R^n） or the Orlicz-Hardy space HФ （R^n）; moreover, the authors establish a new subtle molecular characterization of HФ, L （R^n） associated with L and, as applications, the authors then show that the corresponding fractional integral L^-γ for certain γ∈ E （0,∞） maps HФ, L（R^n） continuously into HФ, L（R^n）, where Ф satisfies the same properties as Ф and is determined by Ф and λ and also that L has a bounded holomorphic functional calculus in HФ, L（R^n）. All these results are new even when Ф（t） = t^p for all t ∈ （0, ∞） and p ∈ （n/（n ＋ θ（L））, 1].展开更多
Let H be a complex infinite dimensional Hilbert space and B(H)be the algebra of all bounded linear operators on H.In this paper,we mainly study the operators that satisfy both a-Weyl's theorem and property(R).Also...Let H be a complex infinite dimensional Hilbert space and B(H)be the algebra of all bounded linear operators on H.In this paper,we mainly study the operators that satisfy both a-Weyl's theorem and property(R).Also,the operators whose functional calculus satisfies the two properties are also explored.We give the features for the operator or its functional calculus for which both a-Weyl's theorem and property(R)hold.展开更多
Let H be a Schroedinger operator on R^n. Under a polynomial decay condition for the kernel of its spectral operator, we show that the Besov spaces and Triebel-Lizorkin spaces associated with H are well defined. We fur...Let H be a Schroedinger operator on R^n. Under a polynomial decay condition for the kernel of its spectral operator, we show that the Besov spaces and Triebel-Lizorkin spaces associated with H are well defined. We further give a Littlewood-Paley characterization of Lp spaces in terms of dyadic functions of H. This generalizes and strengthens the previous result when the heat kernel of H satisfies certain upper Gaussian bound.展开更多
Let A be the matrix associated with an abstract parabolic system in the sense of Shilov or correct system in the sense of Petrovskii. We show that if the spectrum of its symbol is contained in a sector including some ...Let A be the matrix associated with an abstract parabolic system in the sense of Shilov or correct system in the sense of Petrovskii. We show that if the spectrum of its symbol is contained in a sector including some negative real axis, then A generates an analytic regularized semigroup. The corresponding result related to numerical range conditions is also showed. Moreover, these results are applied to matrices of partial differential operators on many function spaces.展开更多
According to the necessary condition of the functional taking the extremum, that is its first variation is equal to zero, the variational problems of the functionals for the undetermined boundary in the calculus of va...According to the necessary condition of the functional taking the extremum, that is its first variation is equal to zero, the variational problems of the functionals for the undetermined boundary in the calculus of variations are researched, the functionals depend on single argument, arbitrary unknown functions and their derivatives of higher orders. A new view point is posed and demonstrated, i.e. when the first variation of the functional is equal to zero, all the variational terms are not independent to each other, and at least one of them is equal to zero. Some theorems and corollaries of the variational problems of the functionals are obtained.展开更多
We study optimal insider control problems,i.e.,optimal control problems of stochastic systemswhere the controller at any time t,in addition to knowledge about the history of the system up to this time,also has additio...We study optimal insider control problems,i.e.,optimal control problems of stochastic systemswhere the controller at any time t,in addition to knowledge about the history of the system up to this time,also has additional information related to a future value of the system.Since this puts the associated controlled systems outside the context of semimartingales,we apply anticipative white noise analysis,including forward integration and Hida-Malliavin calculus to study the problem.Combining this with Donsker delta functionals,we transform the insider control problem into a classical(but parametrised)adapted control system,albeit with a non-classical performance functional.We establish a sufficient and a necessary maximum principle for such systems.Then we apply the results to obtain explicit solutions for some optimal insider portfolio problems in financial markets described by Itô-Lévy processes.Finally,in the Appendix,we give a brief survey of the concepts and results we need from the theory of white noise,forward integrals and Hida-Malliavin calculus.展开更多
The operaton on the n-complex unit sphere under study have three forms: the singular integrals with holomorphic kernels, the bounded and holomorphic Fourier multipliers, and the Cauchy-Dunford bounded and holomorphic ...The operaton on the n-complex unit sphere under study have three forms: the singular integrals with holomorphic kernels, the bounded and holomorphic Fourier multipliers, and the Cauchy-Dunford bounded and holomorphic functional calculus of the radial Dirac operator $D = \sum\nolimits_{k = 1}^n {z_k \frac{\partial }{{\partial _{z_k } }}} $ . The equivalence between the three fom and the strong-type (p,p), 1 <p < ∞, and weak-type (1,1)-boundedness of the operators is proved. The results generalise the work of L. K. Hua, A. Korányli and S. Vagi, W. Rudin and S. Gong on the Cauchy-Szeg?, kemel and the Cauchy singular integral operator.展开更多
A general approach to transference principles for discrete and continuous sequence of operators (semi) groups is described. This allows one to recover the classical transference results of Calderon, Coifman and Weiss ...A general approach to transference principles for discrete and continuous sequence of operators (semi) groups is described. This allows one to recover the classical transference results of Calderon, Coifman and Weiss and of Berkson, Gilleppie and Muhly and the more recent one of the author. The method is applied to derive a new transference principle for (discrete and continuous) the sequence of operators semigroups that need not be grouped. As an application, functional calculus estimates for bounded sequence of operators with at most polynomially growing powers are derived, leading to a new proof of classical results by Peller from 1982. The method allows for a generalization of his results away from Hilbert spaces to -spaces and—involving the concept of γ-boundedness—to general spaces. Analogous results for strongly-continuous one-parameter (semi) groups are presented as well by Markus Haase [1]. Finally, an application is given to singular integrals for one-parameter semigroups.展开更多
Many reduction systems have been presented for implementing functional programming languages. We propose here an extension of a reduction architecture to realize a kind of logic programming——pure Horn clause logic p...Many reduction systems have been presented for implementing functional programming languages. We propose here an extension of a reduction architecture to realize a kind of logic programming——pure Horn clause logic programming.This is an attempt to approach amalgama- tion of the two important programming paradigms.展开更多
We obtain certain time decay and regularity estimates for 3D Schroedinger equation with a potential in the Kato class by using Besov spaces associated with Schroedinger operators.
We prove a complex and a real interpolation theorems on Besov spaces and Triebel-Lizorkin spaces associated with a selfadjoint operator L, without assuming the gradient estimate for its spectral kernel. The result app...We prove a complex and a real interpolation theorems on Besov spaces and Triebel-Lizorkin spaces associated with a selfadjoint operator L, without assuming the gradient estimate for its spectral kernel. The result applies to the cases where L is a uniformly elliptic operator or a Schrdinger operator with electro-magnetic potential.展开更多
The matrix Wiener algebra,W_(N):=M_(N)(W)of order N>0,is the matrix algebra formed by N×N matrices whose entries belong to the classical Wiener algebraWof functions with absolutely convergent Fourier series.A ...The matrix Wiener algebra,W_(N):=M_(N)(W)of order N>0,is the matrix algebra formed by N×N matrices whose entries belong to the classical Wiener algebraWof functions with absolutely convergent Fourier series.A block-Toeplitz matrix T(a)=[A_(i,j)]i,j≥0is a block semi-infinite matrix such that its blocks A_(i,j) are finite matrices of order N,A_(i,j)=A^(r,s) whenever i-j=r-s and its entries are the coefficients of the Fourier expansion of the generator a:T→M_(N)(C).Such a matrix can be regarded as a bounded linear operator acting on the direct sum of N copies of L^(2)(T).We show that exp(T(a))differes from T(exp(a))only in a compact operator with a known bound on its norm.In fact,we prove a slightly more general result:for every entire function f and for every compact operator E,there exists a compact operator F such that f(T(a)+E)=T(f(a))+F.We call these T(a)+E′s matrices,the quasi block-Toeplitz matrices,and we show that via a computation-friendly norm,they form a Banach algebra.Our results generalize and are motivated by some recent results of Dario Andrea Bini,Stefano Massei and Beatrice Meini.展开更多
Let be a Schr?dinger operator on . We show that gradient estimates for the heat kernel of with upper Gaussian bounds imply polynomial decay for the kernels of certain smooth dyadic spectral operators. The latter decay...Let be a Schr?dinger operator on . We show that gradient estimates for the heat kernel of with upper Gaussian bounds imply polynomial decay for the kernels of certain smooth dyadic spectral operators. The latter decay property has been known to play an important role in the Littlewood-Paley theory for and Sobolev spaces. We are able to establish the result by modifying Hebisch and the author’s recent proofs. We give a counterexample in one dimension to show that there exists in the Schwartz class such that the long time gradient heat kernel estimate fails.展开更多
We study the convergence and asymptotic compatibility of higher order collocation methods for nonlocal operators inspired by peridynamics,a nonlocal formulation of continuum mechanics.We prove that the methods are opt...We study the convergence and asymptotic compatibility of higher order collocation methods for nonlocal operators inspired by peridynamics,a nonlocal formulation of continuum mechanics.We prove that the methods are optimally convergent with respect to the polynomial degree of the approximation.A numerical method is said to be asymptotically compatible if the sequence of approximate solutions of the nonlocal problem converges to the solution of the corresponding local problem as the horizon and the grid sizes simultaneously approach zero.We carry out a calibration process via Taylor series expansions and a scaling of the nonlocal operator via a strain energy density argument to ensure that the resulting collocation methods are asymptotically compatible.We fnd that,for polynomial degrees greater than or equal to two,there exists a calibration constant independent of the horizon size and the grid size such that the resulting collocation methods for the nonlocal difusion are asymptotically compatible.We verify these fndings through extensive numerical experiments.展开更多
基金supported by National Natural Science Foundation of China (Grant No. 10871025)Program for Changjiang Scholars and Innovative Research Team in Universities of China
文摘Let L be a linear operator in L^2（R^n） and generate an analytic semigroup {e^-tL}t≥0 with kernel satisfying an upper bound estimate of Poisson type, whose decay is measured by θ（L） ∈ （0, ∞）. Let 4） be a positive, continuous and strictly increasing function on （0, ∞）, which is of strictly critical lower type pФ （n/（n ＋ θ（L））, 1]. Denote by HФ, L（R^n） the Orlicz-Hardy space introduced in Jiang, Yang and Zhou＇s paper in 2009. If Ф is additionally of upper type 1 and subadditive, the authors then show that the Littlewood-Paley g-function gL maps HФ, L（R^n） continuously into LФ（R^n） and, moreover, the authors characterize HФ, L（R^n） in terms of the Littlewood-Paley gλ^＊-function with λ ∈ （n（2/pФ ＋ 1）, ∞）. If Ф is further slightly strengthened to be concave, the authors show that a generalized Riesz transform associated with L is bounded from HФ, L（R^n） to the Orlicz space L^Ф（R^n） or the Orlicz-Hardy space HФ （R^n）; moreover, the authors establish a new subtle molecular characterization of HФ, L （R^n） associated with L and, as applications, the authors then show that the corresponding fractional integral L^-γ for certain γ∈ E （0,∞） maps HФ, L（R^n） continuously into HФ, L（R^n）, where Ф satisfies the same properties as Ф and is determined by Ф and λ and also that L has a bounded holomorphic functional calculus in HФ, L（R^n）. All these results are new even when Ф（t） = t^p for all t ∈ （0, ∞） and p ∈ （n/（n ＋ θ（L））, 1].
基金Supported by the National Natural Science Foundation of China(Grant No.11671201)。
文摘Let H be a complex infinite dimensional Hilbert space and B(H)be the algebra of all bounded linear operators on H.In this paper,we mainly study the operators that satisfy both a-Weyl's theorem and property(R).Also,the operators whose functional calculus satisfies the two properties are also explored.We give the features for the operator or its functional calculus for which both a-Weyl's theorem and property(R)hold.
文摘Let H be a Schroedinger operator on R^n. Under a polynomial decay condition for the kernel of its spectral operator, we show that the Besov spaces and Triebel-Lizorkin spaces associated with H are well defined. We further give a Littlewood-Paley characterization of Lp spaces in terms of dyadic functions of H. This generalizes and strengthens the previous result when the heat kernel of H satisfies certain upper Gaussian bound.
基金This work was supported by the National Natural Science Foundation of China(Grant No. 19971031) Fok Ying Tung Education Foundation, and the Foundation for Excellent Young Teachers of China.
文摘Let A be the matrix associated with an abstract parabolic system in the sense of Shilov or correct system in the sense of Petrovskii. We show that if the spectrum of its symbol is contained in a sector including some negative real axis, then A generates an analytic regularized semigroup. The corresponding result related to numerical range conditions is also showed. Moreover, these results are applied to matrices of partial differential operators on many function spaces.
文摘According to the necessary condition of the functional taking the extremum, that is its first variation is equal to zero, the variational problems of the functionals for the undetermined boundary in the calculus of variations are researched, the functionals depend on single argument, arbitrary unknown functions and their derivatives of higher orders. A new view point is posed and demonstrated, i.e. when the first variation of the functional is equal to zero, all the variational terms are not independent to each other, and at least one of them is equal to zero. Some theorems and corollaries of the variational problems of the functionals are obtained.
文摘We study optimal insider control problems,i.e.,optimal control problems of stochastic systemswhere the controller at any time t,in addition to knowledge about the history of the system up to this time,also has additional information related to a future value of the system.Since this puts the associated controlled systems outside the context of semimartingales,we apply anticipative white noise analysis,including forward integration and Hida-Malliavin calculus to study the problem.Combining this with Donsker delta functionals,we transform the insider control problem into a classical(but parametrised)adapted control system,albeit with a non-classical performance functional.We establish a sufficient and a necessary maximum principle for such systems.Then we apply the results to obtain explicit solutions for some optimal insider portfolio problems in financial markets described by Itô-Lévy processes.Finally,in the Appendix,we give a brief survey of the concepts and results we need from the theory of white noise,forward integrals and Hida-Malliavin calculus.
文摘The operaton on the n-complex unit sphere under study have three forms: the singular integrals with holomorphic kernels, the bounded and holomorphic Fourier multipliers, and the Cauchy-Dunford bounded and holomorphic functional calculus of the radial Dirac operator $D = \sum\nolimits_{k = 1}^n {z_k \frac{\partial }{{\partial _{z_k } }}} $ . The equivalence between the three fom and the strong-type (p,p), 1 <p < ∞, and weak-type (1,1)-boundedness of the operators is proved. The results generalise the work of L. K. Hua, A. Korányli and S. Vagi, W. Rudin and S. Gong on the Cauchy-Szeg?, kemel and the Cauchy singular integral operator.
文摘A general approach to transference principles for discrete and continuous sequence of operators (semi) groups is described. This allows one to recover the classical transference results of Calderon, Coifman and Weiss and of Berkson, Gilleppie and Muhly and the more recent one of the author. The method is applied to derive a new transference principle for (discrete and continuous) the sequence of operators semigroups that need not be grouped. As an application, functional calculus estimates for bounded sequence of operators with at most polynomially growing powers are derived, leading to a new proof of classical results by Peller from 1982. The method allows for a generalization of his results away from Hilbert spaces to -spaces and—involving the concept of γ-boundedness—to general spaces. Analogous results for strongly-continuous one-parameter (semi) groups are presented as well by Markus Haase [1]. Finally, an application is given to singular integrals for one-parameter semigroups.
基金This work was done by the author when he was a visiting researcher in the research group of Prof.Dr.Werner Kluge at Kiel University,supported by the grant of Federal M ster of Science and Technology of Germany.
文摘Many reduction systems have been presented for implementing functional programming languages. We propose here an extension of a reduction architecture to realize a kind of logic programming——pure Horn clause logic programming.This is an attempt to approach amalgama- tion of the two important programming paradigms.
文摘We obtain certain time decay and regularity estimates for 3D Schroedinger equation with a potential in the Kato class by using Besov spaces associated with Schroedinger operators.
文摘We prove a complex and a real interpolation theorems on Besov spaces and Triebel-Lizorkin spaces associated with a selfadjoint operator L, without assuming the gradient estimate for its spectral kernel. The result applies to the cases where L is a uniformly elliptic operator or a Schrdinger operator with electro-magnetic potential.
文摘The matrix Wiener algebra,W_(N):=M_(N)(W)of order N>0,is the matrix algebra formed by N×N matrices whose entries belong to the classical Wiener algebraWof functions with absolutely convergent Fourier series.A block-Toeplitz matrix T(a)=[A_(i,j)]i,j≥0is a block semi-infinite matrix such that its blocks A_(i,j) are finite matrices of order N,A_(i,j)=A^(r,s) whenever i-j=r-s and its entries are the coefficients of the Fourier expansion of the generator a:T→M_(N)(C).Such a matrix can be regarded as a bounded linear operator acting on the direct sum of N copies of L^(2)(T).We show that exp(T(a))differes from T(exp(a))only in a compact operator with a known bound on its norm.In fact,we prove a slightly more general result:for every entire function f and for every compact operator E,there exists a compact operator F such that f(T(a)+E)=T(f(a))+F.We call these T(a)+E′s matrices,the quasi block-Toeplitz matrices,and we show that via a computation-friendly norm,they form a Banach algebra.Our results generalize and are motivated by some recent results of Dario Andrea Bini,Stefano Massei and Beatrice Meini.
文摘Let be a Schr?dinger operator on . We show that gradient estimates for the heat kernel of with upper Gaussian bounds imply polynomial decay for the kernels of certain smooth dyadic spectral operators. The latter decay property has been known to play an important role in the Littlewood-Paley theory for and Sobolev spaces. We are able to establish the result by modifying Hebisch and the author’s recent proofs. We give a counterexample in one dimension to show that there exists in the Schwartz class such that the long time gradient heat kernel estimate fails.
文摘We study the convergence and asymptotic compatibility of higher order collocation methods for nonlocal operators inspired by peridynamics,a nonlocal formulation of continuum mechanics.We prove that the methods are optimally convergent with respect to the polynomial degree of the approximation.A numerical method is said to be asymptotically compatible if the sequence of approximate solutions of the nonlocal problem converges to the solution of the corresponding local problem as the horizon and the grid sizes simultaneously approach zero.We carry out a calibration process via Taylor series expansions and a scaling of the nonlocal operator via a strain energy density argument to ensure that the resulting collocation methods are asymptotically compatible.We fnd that,for polynomial degrees greater than or equal to two,there exists a calibration constant independent of the horizon size and the grid size such that the resulting collocation methods for the nonlocal difusion are asymptotically compatible.We verify these fndings through extensive numerical experiments.
