In this paper,we obtained the extremal function for a weighted singular Trudinger-Moser inequality by blow-up analysis in the Euclidean space R^(2).This extends recent results of Hou(J.Inequal.Appl.,2018)and similar r...In this paper,we obtained the extremal function for a weighted singular Trudinger-Moser inequality by blow-up analysis in the Euclidean space R^(2).This extends recent results of Hou(J.Inequal.Appl.,2018)and similar result was proved by Zhu(Sci.China Math.,2021).展开更多
The theta (t)-type oscillatory singular integral operators has been discussed. With the non-negative locally integrable weighted function, the weighted norm inequality of theta ( t) -type oscillatory singular integral...The theta (t)-type oscillatory singular integral operators has been discussed. With the non-negative locally integrable weighted function, the weighted norm inequality of theta ( t) -type oscillatory singular integral operators is proved, and the weighted function has replaced by action of Hardy-Littlewood maximal operators several times.展开更多
We study boundary value problems for fractional integro-differential equations involving Caputo derivative of order α∈ （n-1, n） in Banach spaces. Existence and uniqueness results of solutions are established by vi...We study boundary value problems for fractional integro-differential equations involving Caputo derivative of order α∈ （n-1, n） in Banach spaces. Existence and uniqueness results of solutions are established by virtue of the Holder＇s inequality, a suitable singular Cronwall＇s inequality and fixed point theorem via a priori estimate method. At last, examples are given to illustrate the results.展开更多
In this article, we give the three-sphere inequalities and three-ball inequalities for the singular elliptic equation div（A∨u） - Vu =0, and the three-ball inequalities on the characteristic plane and the three-cyli...In this article, we give the three-sphere inequalities and three-ball inequalities for the singular elliptic equation div（A∨u） - Vu =0, and the three-ball inequalities on the characteristic plane and the three-cylinder inequalities for the singular parabolic equation Эtu-div（A∨u） ＋ Vu = 0, where the singular potential V belonging to the Kato-Fefferman- Phong＇s class. Some applications are also discussed.展开更多
Here we consider some weighted logarithmic Sobolev inequalities which can be used in the theory of singular Riemanian manifolds.We give the necessary and sufficient conditions such that the 1-dimension weighted logari...Here we consider some weighted logarithmic Sobolev inequalities which can be used in the theory of singular Riemanian manifolds.We give the necessary and sufficient conditions such that the 1-dimension weighted logarithmic Sobolev inequality is true and obtain a new estimate on the entropy.展开更多
The well-known arithmetic-geometric mean inequality for singular values, due to Bhatia and Kittaneh, is one of the most important singular value inequalities for compact operators. The purpose of this study is to give...The well-known arithmetic-geometric mean inequality for singular values, due to Bhatia and Kittaneh, is one of the most important singular value inequalities for compact operators. The purpose of this study is to give new singular value inequalities for compact operators and prove that these inequalities are equivalent to arithmetic-geometric mean inequality, the way by which several future studies could be done.展开更多
@1 Definition 1 Let A=(α<sub>ij</sub>)∈C<sup>n×n</sup>,B=(b<sub>ij</sub>)∈C<sup>n×n</sup>,is nonsingular.The generalizedsingular values of A（relative to B...@1 Definition 1 Let A=(α<sub>ij</sub>)∈C<sup>n×n</sup>,B=(b<sub>ij</sub>)∈C<sup>n×n</sup>,is nonsingular.The generalizedsingular values of A（relative to B）are following determinate nonnegative real numberswhen ||·||<sub>2</sub> denotes the Euclid vector norm,〈n〉={1,2,…,n}.Definition 2 Let A,B∈C<sup>n×n</sup>,if there exist λ∈C and x∈C<sup>n</sup>\{0}。展开更多
We give singular value inequality to compact normal operators, which states that if is compact normal operator on a complex separable Hilbert space, where is the cartesian decomposition of , then Moreover, we give ine...We give singular value inequality to compact normal operators, which states that if is compact normal operator on a complex separable Hilbert space, where is the cartesian decomposition of , then Moreover, we give inequality which asserts that if?is compact normal operator, then .Several inequalities will be proved.展开更多
The well-known arithmetic-geometric mean inequality for singular values, according to Bhatia and Kittaneh, says that if and are compact operators on a complex separable Hilbert space, then Hirzallah has proved that if...The well-known arithmetic-geometric mean inequality for singular values, according to Bhatia and Kittaneh, says that if and are compact operators on a complex separable Hilbert space, then Hirzallah has proved that if are compact operators, then We give inequality which is equivalent to and more general than the above inequalities, which states that if are compact operators,展开更多
In this paper we derive some inequalities for traces and singular values of the quaternion matrices,extend and improve some of the corresponding results appeared in other papers we know.
Let W^(1,n)(R^(n))be the standard Sobolev space.For any T>0 and p>n>2,we denote■Define a norm in W^(1,n)(R^(n))by■where 0≤α<λ_(n,p).Using a rearrangement argument and blow-up analysis,we will prove■c...Let W^(1,n)(R^(n))be the standard Sobolev space.For any T>0 and p>n>2,we denote■Define a norm in W^(1,n)(R^(n))by■where 0≤α<λ_(n,p).Using a rearrangement argument and blow-up analysis,we will prove■can be attained by some function u_(0)∈W^(1,n)(R^(n))∩C^(1)(R^(n))with ||u_(0)||_(n,p)=1,here a_(n)=n■_(n-1)^(1/n-1) and ■_(n-1) is the measure of the unit sphere in R^(n).展开更多
In this paper, using the method of blow-up analysis, we establish a generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularities. Precisely, let(Σ, D) be such a surface∑with divi...In this paper, using the method of blow-up analysis, we establish a generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularities. Precisely, let(Σ, D) be such a surface∑with divisor D =Σ_(i=1)~mβ_(ipi), where β_i >-1 and p_i ∈Σ for i = 1,..., m, and g be a metric representing D.Denote b_0 = 4π(1 + min_(1≤i≤mβ_i). Suppose ψ : Σ→ R is a continuous function with ∫_Σψdv_g ≠0 and define■Then for any α∈ [0, λ_1^(**)(Σ, g)), we have■When b > b0, the integrals■are still finite, but the supremum is infinity. Moreover, we prove that the extremal function for the above inequality exists.展开更多
Let(∑,g)be a compact Riemannian surface,pj∈∑,βj>-1,forj=1,..i,m.Denoteβ=min{0,β1……Bm}.Let H∈C^(0)(∑)be a positivefunction and h(x)=H(x)(dg(x,pj))^(2βj),where dg(x,pj)denotes the geodesic distance between...Let(∑,g)be a compact Riemannian surface,pj∈∑,βj>-1,forj=1,..i,m.Denoteβ=min{0,β1……Bm}.Let H∈C^(0)(∑)be a positivefunction and h(x)=H(x)(dg(x,pj))^(2βj),where dg(x,pj)denotes the geodesic distance between x and p;for each j=1,...,m.In this paper,using a method of blow-up analysis,we prove that the functional J(u)=1/2∫∑|ΔgU|^(2)dV_(g)+8π(1+β)1/volg(∑)∫∑udvg-8π(1+β)log∫_(∑)he^(U)dv_(g)is bounded from below on the Sobolev space w^(1,2)(g).展开更多
The problem of robust H-infinity control for a class of uncertain singular time-delay systems is studied in this paper. A new approach is proposed to describe the relationship between slow and fast subsystems of singu...The problem of robust H-infinity control for a class of uncertain singular time-delay systems is studied in this paper. A new approach is proposed to describe the relationship between slow and fast subsystems of singular time- delay systems, based on which, a sufficient condition is presented for a singular time-delay system to be regular, impulse free and stable with an H-infinity performance. The robust H-infinity control problem is solved and an explicit expression of the desired state-feedback control law is also given. The obtained results are formulated in terms of strict linear matrix inequalities （LMIs） involving no decomposition of system matrices. A numerical example is given to show the effectiveness of the proposed method.展开更多
文摘In this paper,we obtained the extremal function for a weighted singular Trudinger-Moser inequality by blow-up analysis in the Euclidean space R^(2).This extends recent results of Hou(J.Inequal.Appl.,2018)and similar result was proved by Zhu(Sci.China Math.,2021).
基金Foundation item:the Education Commission of Shandong Province(J98P51)
文摘The theta (t)-type oscillatory singular integral operators has been discussed. With the non-negative locally integrable weighted function, the weighted norm inequality of theta ( t) -type oscillatory singular integral operators is proved, and the weighted function has replaced by action of Hardy-Littlewood maximal operators several times.
基金supported by Grant In Aid research fund of Virginia Military Instittue, USA
文摘We study boundary value problems for fractional integro-differential equations involving Caputo derivative of order α∈ （n-1, n） in Banach spaces. Existence and uniqueness results of solutions are established by virtue of the Holder＇s inequality, a suitable singular Cronwall＇s inequality and fixed point theorem via a priori estimate method. At last, examples are given to illustrate the results.
基金supported in part by the NNSF of China (10471069, 10771110)by NSF of Ningbo City (2009A610084)
文摘In this article, we give the three-sphere inequalities and three-ball inequalities for the singular elliptic equation div（A∨u） - Vu =0, and the three-ball inequalities on the characteristic plane and the three-cylinder inequalities for the singular parabolic equation Эtu-div（A∨u） ＋ Vu = 0, where the singular potential V belonging to the Kato-Fefferman- Phong＇s class. Some applications are also discussed.
基金Supported by the National Natural Science Foundation of China(11871436)。
文摘Here we consider some weighted logarithmic Sobolev inequalities which can be used in the theory of singular Riemanian manifolds.We give the necessary and sufficient conditions such that the 1-dimension weighted logarithmic Sobolev inequality is true and obtain a new estimate on the entropy.
文摘The well-known arithmetic-geometric mean inequality for singular values, due to Bhatia and Kittaneh, is one of the most important singular value inequalities for compact operators. The purpose of this study is to give new singular value inequalities for compact operators and prove that these inequalities are equivalent to arithmetic-geometric mean inequality, the way by which several future studies could be done.
文摘@1 Definition 1 Let A=(α<sub>ij</sub>)∈C<sup>n×n</sup>,B=(b<sub>ij</sub>)∈C<sup>n×n</sup>,is nonsingular.The generalizedsingular values of A（relative to B）are following determinate nonnegative real numberswhen ||·||<sub>2</sub> denotes the Euclid vector norm,〈n〉={1,2,…,n}.Definition 2 Let A,B∈C<sup>n×n</sup>,if there exist λ∈C and x∈C<sup>n</sup>\{0}。
文摘We give singular value inequality to compact normal operators, which states that if is compact normal operator on a complex separable Hilbert space, where is the cartesian decomposition of , then Moreover, we give inequality which asserts that if?is compact normal operator, then .Several inequalities will be proved.
文摘The well-known arithmetic-geometric mean inequality for singular values, according to Bhatia and Kittaneh, says that if and are compact operators on a complex separable Hilbert space, then Hirzallah has proved that if are compact operators, then We give inequality which is equivalent to and more general than the above inequalities, which states that if are compact operators,
文摘In this paper we derive some inequalities for traces and singular values of the quaternion matrices,extend and improve some of the corresponding results appeared in other papers we know.
基金supported by National Science Foundation of China(Grant No.12201234)Natural Science Foundation of Anhui Province of China(Grant No.2008085MA07)the Natural Science Foundation of the Education Department of Anhui Province(Grant No.KJ2020A1198).
文摘Let W^(1,n)(R^(n))be the standard Sobolev space.For any T>0 and p>n>2,we denote■Define a norm in W^(1,n)(R^(n))by■where 0≤α<λ_(n,p).Using a rearrangement argument and blow-up analysis,we will prove■can be attained by some function u_(0)∈W^(1,n)(R^(n))∩C^(1)(R^(n))with ||u_(0)||_(n,p)=1,here a_(n)=n■_(n-1)^(1/n-1) and ■_(n-1) is the measure of the unit sphere in R^(n).
基金supported by National Natural Science Foundation of China (Grant No. 11401575)
文摘In this paper, using the method of blow-up analysis, we establish a generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularities. Precisely, let(Σ, D) be such a surface∑with divisor D =Σ_(i=1)~mβ_(ipi), where β_i >-1 and p_i ∈Σ for i = 1,..., m, and g be a metric representing D.Denote b_0 = 4π(1 + min_(1≤i≤mβ_i). Suppose ψ : Σ→ R is a continuous function with ∫_Σψdv_g ≠0 and define■Then for any α∈ [0, λ_1^(**)(Σ, g)), we have■When b > b0, the integrals■are still finite, but the supremum is infinity. Moreover, we prove that the extremal function for the above inequality exists.
基金the National Science Foundation of China(GrantNo.11401575).
文摘Let(∑,g)be a compact Riemannian surface,pj∈∑,βj>-1,forj=1,..i,m.Denoteβ=min{0,β1……Bm}.Let H∈C^(0)(∑)be a positivefunction and h(x)=H(x)(dg(x,pj))^(2βj),where dg(x,pj)denotes the geodesic distance between x and p;for each j=1,...,m.In this paper,using a method of blow-up analysis,we prove that the functional J(u)=1/2∫∑|ΔgU|^(2)dV_(g)+8π(1+β)1/volg(∑)∫∑udvg-8π(1+β)log∫_(∑)he^(U)dv_(g)is bounded from below on the Sobolev space w^(1,2)(g).
基金This work was supported by the National Creative Research Groups Science Foundation of China (No. 60421002) and the New Century 151 Talent Projectof Zhejiang Province.
文摘The problem of robust H-infinity control for a class of uncertain singular time-delay systems is studied in this paper. A new approach is proposed to describe the relationship between slow and fast subsystems of singular time- delay systems, based on which, a sufficient condition is presented for a singular time-delay system to be regular, impulse free and stable with an H-infinity performance. The robust H-infinity control problem is solved and an explicit expression of the desired state-feedback control law is also given. The obtained results are formulated in terms of strict linear matrix inequalities （LMIs） involving no decomposition of system matrices. A numerical example is given to show the effectiveness of the proposed method.
