It is well known that the commutator Tb of the Calderbn-Zygmund singular integral operator is bounded on LP（Rn） for 1 〈 p 〈 ＋∞ if and only if b E BMO [1]. On the other hand, the commutator Tb is bounded from H1...It is well known that the commutator Tb of the Calderbn-Zygmund singular integral operator is bounded on LP（Rn） for 1 〈 p 〈 ＋∞ if and only if b E BMO [1]. On the other hand, the commutator Tb is bounded from H1（Rn） into L1（Rn） only if the function b is a constant [2]. In this article, we will discuss the boundedness of commutator of certain pseudo-differential operators on Hardy spaces H1. Let Tσ be the operators that its symbol is Sσ1,δ with 0 ≤δ〈 1, if b ∈ LMO∞, then, the commutator [b, Tσ] is bounded from H1（Rn） into L1（Rn） and from L∞（Rn） into BMO（Rn）; If [b,Tσ] is bounded from H1（Rn） into L1（Rn） or L1（Rn） into BMO（Rn）, then, b ∈ LMOtoc.展开更多
In this paper, the pseudo-differential operators and the generalized Lax equations in integrable systems are implemented in symbolic software Mathematica. A great deal of differential polynomials which appear in the p...In this paper, the pseudo-differential operators and the generalized Lax equations in integrable systems are implemented in symbolic software Mathematica. A great deal of differential polynomials which appear in the procedure are dealt with by differential characteristic chain method. Using the program, several classical examples are given.展开更多
Consider a pseudo-differential operator T_(a)f(x)=∫_(R^(n))e^(ix,ζ)a(x,ζ)f(ζ)dζwhere the symbol a is in the rough Hormander class L^(∞)S_(ρ)^(m)with m∈R andρ∈[0,1].In this note,when 1≤p≤2,if n(ρ-1)/p and ...Consider a pseudo-differential operator T_(a)f(x)=∫_(R^(n))e^(ix,ζ)a(x,ζ)f(ζ)dζwhere the symbol a is in the rough Hormander class L^(∞)S_(ρ)^(m)with m∈R andρ∈[0,1].In this note,when 1≤p≤2,if n(ρ-1)/p and a∈L^(∞)S_(ρ)^(m),then for any f∈S(R^(n))and x∈R^(n),we prove that M(T_(a)f)(x)≤C(M(|f|^(p))(x))^(1/p) where M is the Hardy-Littlewood maximal operator.Our theorem improves the known results and the bound on m is sharp,in the sense that n(ρ-1)/p can not be replaced by a larger constant.展开更多
In this paper, we establish a sharp function estimate for the multilinear integral operators associated to the pseudo-differential operators. As the application, we obtain the L<sup>p</sup> (1 p norm ...In this paper, we establish a sharp function estimate for the multilinear integral operators associated to the pseudo-differential operators. As the application, we obtain the L<sup>p</sup> (1 p norm inequalities for the multilinear operators.展开更多
1 Main ResultWe consider a kind of multi-pseudo-differential operators, which is introduced by R. Coifman and Y.Meyer [1].Let symbol function σ（x , a, ε）∈C<sup>∞</sup>(R<sup>n</sup>×...1 Main ResultWe consider a kind of multi-pseudo-differential operators, which is introduced by R. Coifman and Y.Meyer [1].Let symbol function σ（x , a, ε）∈C<sup>∞</sup>(R<sup>n</sup>×R<sup>nm</sup>×R<sup>n</sup>) satisfy the following conditions:where x∈R<sup>n</sup>, a = （a<sub>1</sub>,…,a<sub>m</sub>） a-j∈R<sup>n</sup>,ζ∈R<sup>n</sup>, q = （q<sub>1</sub>,…,q<sub>m</sub>）, q<sub>j</sub>∈（Z+）<sup>n</sup> and q =|q<sub>1</sub>|+…+|q<sub>m</sub>|,The multi-pseudo-differential operator with the symbol o（x,α,ζ） is defined as展开更多
In this paper, the authors consider a class of bilinear pseudo-differential operators with symbols of order 0 and type （1, 0） in the sense of HSrmander and use the atomic decompositions of local Hardy spaces to esta...In this paper, the authors consider a class of bilinear pseudo-differential operators with symbols of order 0 and type （1, 0） in the sense of HSrmander and use the atomic decompositions of local Hardy spaces to establish the boundedness of the bilinear pseudo-differential operators and the bilinear singular integral operators on the product of local Hardy spaces.展开更多
In this paper, by discovering a new fact that the Lebesgue boundedness of a class of pseudo- differential operators implies the Sobolev boundedness of another related class of pseudo-differential operators, the author...In this paper, by discovering a new fact that the Lebesgue boundedness of a class of pseudo- differential operators implies the Sobolev boundedness of another related class of pseudo-differential operators, the authors establish the boundedness of pseudo-differential operators with symbols in Sρ,δ^m on Sobolev spaces, where ∈ R, ρ≤ 1 and δ≤ 1. As its applications, the boundedness of commutators generated by pseudo-differential operators on Sobolev and Bessel potential spaces is deduced. Moreover, the boundedness of pseudo-differential operators on Lipschitz spaces is also obtained.展开更多
In this paper,pseudo-differential operators with homogeneous symbol classes associated with the Weinstein transform are introduced.The boundedness of pseudo-differential operators and commutator between two pseudo-dif...In this paper,pseudo-differential operators with homogeneous symbol classes associated with the Weinstein transform are introduced.The boundedness of pseudo-differential operators and commutator between two pseudo-differential operators on H_(α,2)^(r) are proven with the help of the Weinstein transform technique.展开更多
This article is concerned with the study of pseudo-differential operators associated with fractional Hankel transform. The product of two fractional pseudo-differential operators is defined and investigated its basic ...This article is concerned with the study of pseudo-differential operators associated with fractional Hankel transform. The product of two fractional pseudo-differential operators is defined and investigated its basic properties on some function space. It is shown that the pseudo-differential operators and their products are bounded in Sobolev type spaces. Particular cases are discussed.展开更多
Let R be a ring with a derivation 5 and R（（x-1;5）） denote the pseudo- differential operator ring over R. We study the relations between the set of annihilators in R and the set of annihilators in R（（x-1; 5））. ...Let R be a ring with a derivation 5 and R（（x-1;5）） denote the pseudo- differential operator ring over R. We study the relations between the set of annihilators in R and the set of annihilators in R（（x-1; 5））. Among applications, it is shown that for an Armendariz ring R of pseudo-differential operator type, the ring R（（x-1; 5）） is Baer （resp., quasi-Baer, PP, right zip） if and only if R is a Baer （resp., quasi-Baer, PP, right zip） ring. For a 5-weakly rigid ring R, R（（x-1;5）） is a left p.q.-Baer ring if and only if R is left p.q.-Baer and every countable subset of left semicentral idempotents of R has a generalized countable join in R.展开更多
Pseudo-differential operators(PDO)Q(x,L_(a,x))and Q(x,L_(a,x))involving the index Whittaker transform are defined.Estimates for these operators in Hilbert space L_(2)^(a)(R+;m_(a)(x)dx)are obtained.A symbol classΩis ...Pseudo-differential operators(PDO)Q(x,L_(a,x))and Q(x,L_(a,x))involving the index Whittaker transform are defined.Estimates for these operators in Hilbert space L_(2)^(a)(R+;m_(a)(x)dx)are obtained.A symbol classΩis introduced.Later product and commutators for the PDO are investigated and their boundedness results are discussed.展开更多
The purpose of this paper is to define a new symbol classand discuss the theory of two different pseudo-differential operators(p.d.o.)involving Fourier–Jacobi transform associated with a single symbol in.We also deri...The purpose of this paper is to define a new symbol classand discuss the theory of two different pseudo-differential operators(p.d.o.)involving Fourier–Jacobi transform associated with a single symbol in.We also derive boundedness results for p.d.o.’s in Sobolev type space.Anewpseudo-differential operator is developed using the product of symbols.Finally,norm inequality for commutators between two pseudo-differential operators is obtained.展开更多
In this paper,the boundedness from Lebesgue space to Orlicz space of certain Toeplitz type operator related to the fractional and pseudo-differential operators is obtained.
The aim of this paper is to establish the boundedness of bilinear pseudodifferential operator T_(σ) and its commutator[b_(1),b_(2),T_(σ)]generated by T_(σ) and b_(1),b_(2) BMO(R^(n))on generalized fractional weight...The aim of this paper is to establish the boundedness of bilinear pseudodifferential operator T_(σ) and its commutator[b_(1),b_(2),T_(σ)]generated by T_(σ) and b_(1),b_(2) BMO(R^(n))on generalized fractional weighted Morrey spaces L^(p,η,φ)(w).Under assumption that a weight satisfies a certain condition,the authors prove that Ts is bounded from products of spaces L^(p1,η1,φ)(w1)L^(p2,η2,φ)(w2)into spaces L^(p,η,φ)(w),where w=(w_(1),w_(2)) A_(P),P=(p1,p2),η=η1+η2 and 1/p=1/p_(1)+1/p_(2) with p_(1),p_(2)(1,∞).Furthermore,the authors show that the[b1,b2,T_(σ)]is bounded from products of generalized fractional Morrey spaces L^(p1,η1,φ)(R^(n))L^(p2,η2,φ)(R^(n))into L^(p,η,φ)(R^(n)).As corollaries,the boundedness of the T_(σ) and[b_(1),b_(2),T_(σ)]on generalized weighted Morrey spaces L^(p,φ)(w)and on generalized Morrey spaces L^(p,φ)(R^(n))is also obtained.展开更多
In this paper,we consider the(q,r)boundedness ofthe pseudo-differential operators with the amplitude a∈L^(P)S_(p)^(m)(p≥1,m∈R,0≤p≤1).When 0<r≤∞,1≤p,q≤∞,r≤p,1/r≤1/p+1/q,we provethat if m<(n(p-1))/min(...In this paper,we consider the(q,r)boundedness ofthe pseudo-differential operators with the amplitude a∈L^(P)S_(p)^(m)(p≥1,m∈R,0≤p≤1).When 0<r≤∞,1≤p,q≤∞,r≤p,1/r≤1/p+1/q,we provethat if m<(n(p-1))/min(min{2,p,q})-np((1/p)+(1/q)-(1/r))then for any a∈L^(p)S_(p)^(m),the pseudo-differential operator T_(a)is bounded from L^(q)to L^(r).It is a generalization and improvement of the known theorems and in general the conditions on r,m are sharp.展开更多
Aircraft assembly is characterized by stringent precedence constraints,limited resource availability,spatial restrictions,and a high degree of manual intervention.These factors lead to considerable variability in oper...Aircraft assembly is characterized by stringent precedence constraints,limited resource availability,spatial restrictions,and a high degree of manual intervention.These factors lead to considerable variability in operator workloads and significantly increase the complexity of scheduling.To address this challenge,this study investigates the Aircraft Pulsating Assembly Line Scheduling Problem(APALSP)under skilled operator allocation,with the objective of minimizing assembly completion time.A mathematical model considering skilled operator allocation is developed,and a Q-Learning improved Particle Swarm Optimization algorithm(QLPSO)is proposed.In the algorithm design,a reverse scheduling strategy is adopted to effectively manage large-scale precedence constraints.Moreover,a reverse sequence encoding method is introduced to generate operation sequences,while a time decoding mechanism is employed to determine completion times.The problem is further reformulated as a Markov Decision Process(MDP)with explicitly defined state and action spaces.Within QLPSO,the Q-learning mechanism adaptively adjusts inertia weights and learning factors,thereby achieving a balance between exploration capability and convergence performance.To validate the effectiveness of the proposed approach,extensive computational experiments are conducted on benchmark instances of different scales,including small,medium,large,and ultra-large cases.The results demonstrate that QLPSO consistently delivers stable and high-quality solutions across all scenarios.In ultra-large-scale instances,it improves the best solution by 25.2%compared with the Genetic Algorithm(GA)and enhances the average solution by 16.9%over the Q-learning algorithm,showing clear advantages over the comparative methods.These findings not only confirm the effectiveness of the proposed algorithm but also provide valuable theoretical references and practical guidance for the intelligent scheduling optimization of aircraft pulsating assembly lines.展开更多
Many mathematicians pay their great attention to the study of the concept of derivatives for functions defined on locally compact Vilenkin groups. In this paper, this topic is investigated by virtue of so-called pseud...Many mathematicians pay their great attention to the study of the concept of derivatives for functions defined on locally compact Vilenkin groups. In this paper, this topic is investigated by virtue of so-called pseudo-differential operators. We give the definitions of derivatives and integrals of functions defined on locally compact Vilenkingroups and study their properties, then give an application example.展开更多
The aim of the present paper is to study 2-complex symmetric bounded weighted composition operators on the Fock space of C^(N) with the conjugations J and J_(t,A,b) defined by ■ respectively,where k(z_(1),...,z_N)=(...The aim of the present paper is to study 2-complex symmetric bounded weighted composition operators on the Fock space of C^(N) with the conjugations J and J_(t,A,b) defined by ■ respectively,where k(z_(1),...,z_N)=(■,...,■),t∈C,b∈C^(N) and A is a linear operator on C^(N).An example of 2-complex symmetric bounded weighted composition operator with the conjugation J_(t,A,b) is given.展开更多
基金supported by the National Science Foundation of China NSFC(11161044,11131005)
文摘It is well known that the commutator Tb of the Calderbn-Zygmund singular integral operator is bounded on LP（Rn） for 1 〈 p 〈 ＋∞ if and only if b E BMO [1]. On the other hand, the commutator Tb is bounded from H1（Rn） into L1（Rn） only if the function b is a constant [2]. In this article, we will discuss the boundedness of commutator of certain pseudo-differential operators on Hardy spaces H1. Let Tσ be the operators that its symbol is Sσ1,δ with 0 ≤δ〈 1, if b ∈ LMO∞, then, the commutator [b, Tσ] is bounded from H1（Rn） into L1（Rn） and from L∞（Rn） into BMO（Rn）; If [b,Tσ] is bounded from H1（Rn） into L1（Rn） or L1（Rn） into BMO（Rn）, then, b ∈ LMOtoc.
基金National Natural Science Foundation of China under Grant Nos.10371070 and 10671121the Foundation of Shanghai Education Committee for Shanghai Prospective Excellent Young Teachers,the Youth Foundation of Shanghai Education Committee,and Magnolia Grant of Shanghai Sciences and Technology Committee
文摘Some general formulas in the Sato theory related to the nonisospectral KP and mKP hierarchies are derived for simplifying calculations.
基金The project supported by National Natural Science Foundation of China under Grant No.10401021
文摘In this paper, the pseudo-differential operators and the generalized Lax equations in integrable systems are implemented in symbolic software Mathematica. A great deal of differential polynomials which appear in the procedure are dealt with by differential characteristic chain method. Using the program, several classical examples are given.
基金Supported by the National Natural Science Foundation of China(11871436,12071437)。
文摘Consider a pseudo-differential operator T_(a)f(x)=∫_(R^(n))e^(ix,ζ)a(x,ζ)f(ζ)dζwhere the symbol a is in the rough Hormander class L^(∞)S_(ρ)^(m)with m∈R andρ∈[0,1].In this note,when 1≤p≤2,if n(ρ-1)/p and a∈L^(∞)S_(ρ)^(m),then for any f∈S(R^(n))and x∈R^(n),we prove that M(T_(a)f)(x)≤C(M(|f|^(p))(x))^(1/p) where M is the Hardy-Littlewood maximal operator.Our theorem improves the known results and the bound on m is sharp,in the sense that n(ρ-1)/p can not be replaced by a larger constant.
文摘In this paper, we establish a sharp function estimate for the multilinear integral operators associated to the pseudo-differential operators. As the application, we obtain the L<sup>p</sup> (1 p norm inequalities for the multilinear operators.
文摘1 Main ResultWe consider a kind of multi-pseudo-differential operators, which is introduced by R. Coifman and Y.Meyer [1].Let symbol function σ（x , a, ε）∈C<sup>∞</sup>(R<sup>n</sup>×R<sup>nm</sup>×R<sup>n</sup>) satisfy the following conditions:where x∈R<sup>n</sup>, a = （a<sub>1</sub>,…,a<sub>m</sub>） a-j∈R<sup>n</sup>,ζ∈R<sup>n</sup>, q = （q<sub>1</sub>,…,q<sub>m</sub>）, q<sub>j</sub>∈（Z+）<sup>n</sup> and q =|q<sub>1</sub>|+…+|q<sub>m</sub>|,The multi-pseudo-differential operator with the symbol o（x,α,ζ） is defined as
基金the National Natural Science Foundation of China (Grant No. 10571014)the Doctoral Programme Foundation of Institution of Higher Education of China (Grant No. 20040027001)
文摘In this paper, the boundedness of commutators generated by pseudo-differential operators and BMO functions is discussed on Lebesgue spaces.
基金supported by National Natural Science Foundation of China(Grant No.10861010)
文摘In this paper, the authors consider a class of bilinear pseudo-differential operators with symbols of order 0 and type （1, 0） in the sense of HSrmander and use the atomic decompositions of local Hardy spaces to establish the boundedness of the bilinear pseudo-differential operators and the bilinear singular integral operators on the product of local Hardy spaces.
基金Supported by National Natural Science Foundation of China (Grant No. 10871024)
文摘In this paper, by discovering a new fact that the Lebesgue boundedness of a class of pseudo- differential operators implies the Sobolev boundedness of another related class of pseudo-differential operators, the authors establish the boundedness of pseudo-differential operators with symbols in Sρ,δ^m on Sobolev spaces, where ∈ R, ρ≤ 1 and δ≤ 1. As its applications, the boundedness of commutators generated by pseudo-differential operators on Sobolev and Bessel potential spaces is deduced. Moreover, the boundedness of pseudo-differential operators on Lipschitz spaces is also obtained.
基金Supported by SERB MATRICS(Grant No.MTR2021/000266)。
文摘In this paper,pseudo-differential operators with homogeneous symbol classes associated with the Weinstein transform are introduced.The boundedness of pseudo-differential operators and commutator between two pseudo-differential operators on H_(α,2)^(r) are proven with the help of the Weinstein transform technique.
基金supported by CSIR,New Delhi(Grant No.25(240)/15/EMR-Ⅱ)
文摘This article is concerned with the study of pseudo-differential operators associated with fractional Hankel transform. The product of two fractional pseudo-differential operators is defined and investigated its basic properties on some function space. It is shown that the pseudo-differential operators and their products are bounded in Sobolev type spaces. Particular cases are discussed.
文摘Let R be a ring with a derivation 5 and R（（x-1;5）） denote the pseudo- differential operator ring over R. We study the relations between the set of annihilators in R and the set of annihilators in R（（x-1; 5））. Among applications, it is shown that for an Armendariz ring R of pseudo-differential operator type, the ring R（（x-1; 5）） is Baer （resp., quasi-Baer, PP, right zip） if and only if R is a Baer （resp., quasi-Baer, PP, right zip） ring. For a 5-weakly rigid ring R, R（（x-1;5）） is a left p.q.-Baer ring if and only if R is left p.q.-Baer and every countable subset of left semicentral idempotents of R has a generalized countable join in R.
基金supported by Science and Engineering Research Board,Government of India,under Grant No.EMR/2016/005141。
文摘Pseudo-differential operators(PDO)Q(x,L_(a,x))and Q(x,L_(a,x))involving the index Whittaker transform are defined.Estimates for these operators in Hilbert space L_(2)^(a)(R+;m_(a)(x)dx)are obtained.A symbol classΩis introduced.Later product and commutators for the PDO are investigated and their boundedness results are discussed.
文摘The purpose of this paper is to define a new symbol classand discuss the theory of two different pseudo-differential operators(p.d.o.)involving Fourier–Jacobi transform associated with a single symbol in.We also derive boundedness results for p.d.o.’s in Sobolev type space.Anewpseudo-differential operator is developed using the product of symbols.Finally,norm inequality for commutators between two pseudo-differential operators is obtained.
基金supported by the National Natural Science Foundation of China(Grant No.11901126)the Scientific Research Funds of Hunan Provincial Education Department.(Grant No.19B509).
文摘In this paper,the boundedness from Lebesgue space to Orlicz space of certain Toeplitz type operator related to the fractional and pseudo-differential operators is obtained.
基金supported by the National Natural Science Foundation of China(Grant No.12201500)the Science Foundation for Youths of Gansu Province(Grant No.22JR5RA173)the Young Teachers’Scientific Research Ability Promotion Project of Northwest Normal University(Grant No.NWNU-LKQN2020-07).
文摘The aim of this paper is to establish the boundedness of bilinear pseudodifferential operator T_(σ) and its commutator[b_(1),b_(2),T_(σ)]generated by T_(σ) and b_(1),b_(2) BMO(R^(n))on generalized fractional weighted Morrey spaces L^(p,η,φ)(w).Under assumption that a weight satisfies a certain condition,the authors prove that Ts is bounded from products of spaces L^(p1,η1,φ)(w1)L^(p2,η2,φ)(w2)into spaces L^(p,η,φ)(w),where w=(w_(1),w_(2)) A_(P),P=(p1,p2),η=η1+η2 and 1/p=1/p_(1)+1/p_(2) with p_(1),p_(2)(1,∞).Furthermore,the authors show that the[b1,b2,T_(σ)]is bounded from products of generalized fractional Morrey spaces L^(p1,η1,φ)(R^(n))L^(p2,η2,φ)(R^(n))into L^(p,η,φ)(R^(n)).As corollaries,the boundedness of the T_(σ) and[b_(1),b_(2),T_(σ)]on generalized weighted Morrey spaces L^(p,φ)(w)and on generalized Morrey spaces L^(p,φ)(R^(n))is also obtained.
文摘In this paper,we consider the(q,r)boundedness ofthe pseudo-differential operators with the amplitude a∈L^(P)S_(p)^(m)(p≥1,m∈R,0≤p≤1).When 0<r≤∞,1≤p,q≤∞,r≤p,1/r≤1/p+1/q,we provethat if m<(n(p-1))/min(min{2,p,q})-np((1/p)+(1/q)-(1/r))then for any a∈L^(p)S_(p)^(m),the pseudo-differential operator T_(a)is bounded from L^(q)to L^(r).It is a generalization and improvement of the known theorems and in general the conditions on r,m are sharp.
基金supported by the National Natural Science Foundation of China(Grant No.52475543)Natural Science Foundation of Henan(Grant No.252300421101)+1 种基金Henan Province University Science and Technology Innovation Talent Support Plan(Grant No.24HASTIT048)Science and Technology Innovation Team Project of Zhengzhou University of Light Industry(Grant No.23XNKJTD0101).
文摘Aircraft assembly is characterized by stringent precedence constraints,limited resource availability,spatial restrictions,and a high degree of manual intervention.These factors lead to considerable variability in operator workloads and significantly increase the complexity of scheduling.To address this challenge,this study investigates the Aircraft Pulsating Assembly Line Scheduling Problem(APALSP)under skilled operator allocation,with the objective of minimizing assembly completion time.A mathematical model considering skilled operator allocation is developed,and a Q-Learning improved Particle Swarm Optimization algorithm(QLPSO)is proposed.In the algorithm design,a reverse scheduling strategy is adopted to effectively manage large-scale precedence constraints.Moreover,a reverse sequence encoding method is introduced to generate operation sequences,while a time decoding mechanism is employed to determine completion times.The problem is further reformulated as a Markov Decision Process(MDP)with explicitly defined state and action spaces.Within QLPSO,the Q-learning mechanism adaptively adjusts inertia weights and learning factors,thereby achieving a balance between exploration capability and convergence performance.To validate the effectiveness of the proposed approach,extensive computational experiments are conducted on benchmark instances of different scales,including small,medium,large,and ultra-large cases.The results demonstrate that QLPSO consistently delivers stable and high-quality solutions across all scenarios.In ultra-large-scale instances,it improves the best solution by 25.2%compared with the Genetic Algorithm(GA)and enhances the average solution by 16.9%over the Q-learning algorithm,showing clear advantages over the comparative methods.These findings not only confirm the effectiveness of the proposed algorithm but also provide valuable theoretical references and practical guidance for the intelligent scheduling optimization of aircraft pulsating assembly lines.
基金Project supported by the National Natural Science Foundation of China
文摘Many mathematicians pay their great attention to the study of the concept of derivatives for functions defined on locally compact Vilenkin groups. In this paper, this topic is investigated by virtue of so-called pseudo-differential operators. We give the definitions of derivatives and integrals of functions defined on locally compact Vilenkingroups and study their properties, then give an application example.
基金Supported by Sichuan Science and Technology Program (No.2022ZYD0010)。
文摘The aim of the present paper is to study 2-complex symmetric bounded weighted composition operators on the Fock space of C^(N) with the conjugations J and J_(t,A,b) defined by ■ respectively,where k(z_(1),...,z_N)=(■,...,■),t∈C,b∈C^(N) and A is a linear operator on C^(N).An example of 2-complex symmetric bounded weighted composition operator with the conjugation J_(t,A,b) is given.
