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
The authors in the paper proved that if Ω is homogeneous of degree zero and satisfies some certain logarithmic type Lipschitz condition,then the fractional type Marcinkiewicz Integral μ Ω,α is an operator of type ...The authors in the paper proved that if Ω is homogeneous of degree zero and satisfies some certain logarithmic type Lipschitz condition,then the fractional type Marcinkiewicz Integral μ Ω,α is an operator of type （H˙ K n（1-1/q 1 ）,p q 1 ,˙ K n（1-1/q 1 ）,p q 2 ） and of type （H 1 （R n ）,L n/（n-α） ）.展开更多
We obtain weak type （1, q） inequalities for fractional integral operators on generalized non-homogeneous Morrey spaces. The proofs use some properties of maximal operators. Our results are closely related to the str...We obtain weak type （1, q） inequalities for fractional integral operators on generalized non-homogeneous Morrey spaces. The proofs use some properties of maximal operators. Our results are closely related to the strong type inequalities in [13, 14, 15].展开更多
Most of the existing multivariable grey models are based on the 1-order derivative and 1-order accumulation, which makes the parameters unable to be adjusted according to the data characteristics of the actual problem...Most of the existing multivariable grey models are based on the 1-order derivative and 1-order accumulation, which makes the parameters unable to be adjusted according to the data characteristics of the actual problems. The results about fractional derivative multivariable grey models are very few at present. In this paper, a multivariable Caputo fractional derivative grey model with convolution integral CFGMC(q, N) is proposed. First, the Caputo fractional difference is used to discretize the model, and the least square method is used to solve the parameters. The orders of accumulations and differential equations are determined by using particle swarm optimization(PSO). Then, the analytical solution of the model is obtained by using the Laplace transform, and the convergence and divergence of series in analytical solutions are also discussed. Finally, the CFGMC(q, N) model is used to predict the municipal solid waste(MSW). Compared with other competition models, the model has the best prediction effect. This study enriches the model form of the multivariable grey model, expands the scope of application, and provides a new idea for the development of fractional derivative grey model.展开更多
In this paper, a three dimensional matrix valued rational interpolant (TGMRI) is first constructed by making use of the generalized inverse of matrices. The interpolants are of the Thiele type branched continued fra...In this paper, a three dimensional matrix valued rational interpolant (TGMRI) is first constructed by making use of the generalized inverse of matrices. The interpolants are of the Thiele type branched continued fraction form, with matrix numerator and scalar denominator. Some properties of TGMRI are given. An efficient recursive algorithm is proposed. The results in the paper can be extend to n variable.展开更多
In this paper, we extend the well-studied fractional Brownian motion of Riemann-Liouville type to the multivariate case, and the corresponding processes are called operator fractional Brownian motions of Riemann-Liouv...In this paper, we extend the well-studied fractional Brownian motion of Riemann-Liouville type to the multivariate case, and the corresponding processes are called operator fractional Brownian motions of Riemann-Liouville type. We also provide two results on approximation to operator fractional Brownian motions of Riemann-Liouville type. The first approximation is based on a Poisson process, and the second one is based on a sequence of I.I.D. random variables.展开更多
We present Alikhanov linearized Galerkin methods for solving the nonlinear time fractional Schrödinger equations.Unconditionally optimal estimates of the fully-discrete scheme are obtained by using the fractional...We present Alikhanov linearized Galerkin methods for solving the nonlinear time fractional Schrödinger equations.Unconditionally optimal estimates of the fully-discrete scheme are obtained by using the fractional time-spatial splitting argument.The convergence results indicate that the error estimates hold without any spatial-temporal stepsize restrictions.Numerical experiments are done to verify the theoretical results.展开更多
Similar to the property of a linear Calderdn-Zygmund operator, a linear fractional type operator Is associated with a BMO function b fails to satisfy the continuity from the Hardy space Hp into Lp for p ≤ 1. Thus, an...Similar to the property of a linear Calderdn-Zygmund operator, a linear fractional type operator Is associated with a BMO function b fails to satisfy the continuity from the Hardy space Hp into Lp for p ≤ 1. Thus, an alternative result was given by Y. Ding, S. Lu and P. Zhang, they proved that [b,Iα] is continuous from an atomic Hardy space Hp b into Lp, where Hp b is a subspace of the Hardy space Hp for n/（n ＋ 1） 〈 p ≤ 1. In this paper, we study the commutators of multilinear fractional type operators on product of certain Hardy spaces. The endpoint （Hp1 b1 ×... × HP2, Lp） boundedness for multilinear fractional type operators is obtained. We also give the boundedness for the commutators of multilinear Calderdn-Zygmund operators and multilinear fractional type operators on product of certain Hardy spaces when b ∈ （Lipβ）m（Rn）.展开更多
This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of L1-Galerkin finite element methods.The analysis of L1 methods for time-fractional nonlinear problems is li...This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of L1-Galerkin finite element methods.The analysis of L1 methods for time-fractional nonlinear problems is limited mainly due to the lack of a fundamental Gronwall type inequality.In this paper,we establish such a fundamental inequality for the L1 approximation to the Caputo fractional derivative.In terms of the Gronwall type inequality,we provide optimal error estimates of several fully discrete linearized Galerkin finite element methods for nonlinear problems.The theoretical results are illustrated by applying our proposed methods to the time fractional nonlinear Huxley equation and time fractional Fisher equation.展开更多
We investigate the boundedness of singular and fractional integral operators on generalized Hardy spaces defined on spaces of homogeneous type, which are preduals of Campanato spaces with variable growth condition. To...We investigate the boundedness of singular and fractional integral operators on generalized Hardy spaces defined on spaces of homogeneous type, which are preduals of Campanato spaces with variable growth condition. To do this we introduce molecules with variable growth condition. Our results are new even for R^n case.展开更多
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 National Natural Science Foundation of China (Grant Nos. 10861010 10871024)
文摘The authors in the paper proved that if Ω is homogeneous of degree zero and satisfies some certain logarithmic type Lipschitz condition,then the fractional type Marcinkiewicz Integral μ Ω,α is an operator of type （H˙ K n（1-1/q 1 ）,p q 1 ,˙ K n（1-1/q 1 ）,p q 2 ） and of type （H 1 （R n ）,L n/（n-α） ）.
基金Supported by Fundamental Research Program 2011-2012
文摘We obtain weak type （1, q） inequalities for fractional integral operators on generalized non-homogeneous Morrey spaces. The proofs use some properties of maximal operators. Our results are closely related to the strong type inequalities in [13, 14, 15].
基金supported by the National Natural Science Foundation of China (51479151,61403288)。
文摘Most of the existing multivariable grey models are based on the 1-order derivative and 1-order accumulation, which makes the parameters unable to be adjusted according to the data characteristics of the actual problems. The results about fractional derivative multivariable grey models are very few at present. In this paper, a multivariable Caputo fractional derivative grey model with convolution integral CFGMC(q, N) is proposed. First, the Caputo fractional difference is used to discretize the model, and the least square method is used to solve the parameters. The orders of accumulations and differential equations are determined by using particle swarm optimization(PSO). Then, the analytical solution of the model is obtained by using the Laplace transform, and the convergence and divergence of series in analytical solutions are also discussed. Finally, the CFGMC(q, N) model is used to predict the municipal solid waste(MSW). Compared with other competition models, the model has the best prediction effect. This study enriches the model form of the multivariable grey model, expands the scope of application, and provides a new idea for the development of fractional derivative grey model.
文摘In this paper, a three dimensional matrix valued rational interpolant (TGMRI) is first constructed by making use of the generalized inverse of matrices. The interpolants are of the Thiele type branched continued fraction form, with matrix numerator and scalar denominator. Some properties of TGMRI are given. An efficient recursive algorithm is proposed. The results in the paper can be extend to n variable.
基金Supported by National Natural Science Foundation of China(Grant No.11126343)
文摘In this paper, we extend the well-studied fractional Brownian motion of Riemann-Liouville type to the multivariate case, and the corresponding processes are called operator fractional Brownian motions of Riemann-Liouville type. We also provide two results on approximation to operator fractional Brownian motions of Riemann-Liouville type. The first approximation is based on a Poisson process, and the second one is based on a sequence of I.I.D. random variables.
基金supported by the NSFC (No.12001067)by the Natural Science Foundation of Chongqing,China (No.cstc2019jcyj-bshX0038)by the China Postdoctoral Science Foundation funded Project No.2019M653333.
文摘We present Alikhanov linearized Galerkin methods for solving the nonlinear time fractional Schrödinger equations.Unconditionally optimal estimates of the fully-discrete scheme are obtained by using the fractional time-spatial splitting argument.The convergence results indicate that the error estimates hold without any spatial-temporal stepsize restrictions.Numerical experiments are done to verify the theoretical results.
基金Acknowledgements The authors want to express their sincerely thanks to the referees for their valuable remarks and suggestions which made this paper more readable. This work was supported partly by the National Natural Science Foundation of China （Grant No. 11471041）, the Fundamental Research Funds for the Central Universities （No. 2012CXQT09）, and the Program for New Century Excellent Talents in University （NCET-13-0065）.
文摘Similar to the property of a linear Calderdn-Zygmund operator, a linear fractional type operator Is associated with a BMO function b fails to satisfy the continuity from the Hardy space Hp into Lp for p ≤ 1. Thus, an alternative result was given by Y. Ding, S. Lu and P. Zhang, they proved that [b,Iα] is continuous from an atomic Hardy space Hp b into Lp, where Hp b is a subspace of the Hardy space Hp for n/（n ＋ 1） 〈 p ≤ 1. In this paper, we study the commutators of multilinear fractional type operators on product of certain Hardy spaces. The endpoint （Hp1 b1 ×... × HP2, Lp） boundedness for multilinear fractional type operators is obtained. We also give the boundedness for the commutators of multilinear Calderdn-Zygmund operators and multilinear fractional type operators on product of certain Hardy spaces when b ∈ （Lipβ）m（Rn）.
基金This work is supported by NSFC(Grant Nos.11771035,11771162,11571128,61473126,91430216,91530204,11372354 and U1530401),a grant from the RGC of HK 11300517,China(Project No.CityU 11302915),China Postdoctoral Science Foundation under grant No.2016M602273,a grant DRA2015518 from 333 High-level Personal Training Project of Jiangsu Province,and the USA National Science Foundation grant DMS-1315259the USA Air Force Office of Scientific Research grant FA9550-15-1-0001.Jiwei Zhang also thanks the hospitality of Hong Kong City University during the period of his visiting.
文摘This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of L1-Galerkin finite element methods.The analysis of L1 methods for time-fractional nonlinear problems is limited mainly due to the lack of a fundamental Gronwall type inequality.In this paper,we establish such a fundamental inequality for the L1 approximation to the Caputo fractional derivative.In terms of the Gronwall type inequality,we provide optimal error estimates of several fully discrete linearized Galerkin finite element methods for nonlinear problems.The theoretical results are illustrated by applying our proposed methods to the time fractional nonlinear Huxley equation and time fractional Fisher equation.
基金supported by Grant-in-Aid for Scientific Research (B) (Grant No. 15H03621), Japan Society for the Promotion of Science
文摘We investigate the boundedness of singular and fractional integral operators on generalized Hardy spaces defined on spaces of homogeneous type, which are preduals of Campanato spaces with variable growth condition. To do this we introduce molecules with variable growth condition. Our results are new even for R^n case.
