Microlocal defect measures have been used to study the energy densities of scmilinear wave equations with oscillatory initial data.An "open problem" in literature is solved.
A commutative Noetherian ring R is called a regular Noetherian ring with pure dimension n, if for any maximal ideal m of R, gl.dimR_m=n, where R_m is the localization of R at the maximal ideal m. It is well known that...A commutative Noetherian ring R is called a regular Noetherian ring with pure dimension n, if for any maximal ideal m of R, gl.dimR_m=n, where R_m is the localization of R at the maximal ideal m. It is well known that if R is a finitely generated commutative algebra over some field, R is integral and gl. dimR【∞, then R is a regular Noetherian ring with pure dimension. Let D(V) be the ring of differential operators over the non-singular n-dimensional irreducible algebraic variety V. Then gr(D(V)) is a展开更多
In this paper, we develop semi-classical analysis on H-type groups. We define semi-classical pseudodifferential operators, prove the boundedness of their action on square integrable functions and develop a symbolic ca...In this paper, we develop semi-classical analysis on H-type groups. We define semi-classical pseudodifferential operators, prove the boundedness of their action on square integrable functions and develop a symbolic calculus. Then, we define the semi-classical measures of bounded families of square integrable functions which consist of a pair formed by a measure defined on the product of the group and its unitary dual, and by a field of trace class positive operators acting on the Hilbert spaces of the representations. We illustrate the theory by analyzing examples, which show in particular that this semi-classical analysis takes into account the finite-dimensional representations of the group, even though they are negligible with respect to the Plancherel measure.展开更多
In this paper, the author applies the Egorov Theorem for paradifferential operators established by the theory of para-Fourier integral operators to reduce microlocally paradifferential operators with the principal typ...In this paper, the author applies the Egorov Theorem for paradifferential operators established by the theory of para-Fourier integral operators to reduce microlocally paradifferential operators with the principal type. This reduction is used to study the propagation of singularities for solutions to nonlinear differential equations in the lower frequency range.展开更多
In this note, we use the so-called microlocal energy method to give a characterization of the Gevrey-Sobolev wave front set WF<sub>(</sub>H<sub>T,σ</sub><sup>S</sup> （u）, which w...In this note, we use the so-called microlocal energy method to give a characterization of the Gevrey-Sobolev wave front set WF<sub>(</sub>H<sub>T,σ</sub><sup>S</sup> （u）, which will be useful in the study of non-linear microlocal analysis in Gevrey classes.展开更多
In this paper we generalize the global Sobolev inequality introduced by Klainerman in studying wave equation to the hyperbolic system case.We obtain several decay estimates of solutions of a hyperbolic system of first...In this paper we generalize the global Sobolev inequality introduced by Klainerman in studying wave equation to the hyperbolic system case.We obtain several decay estimates of solutions of a hyperbolic system of first order by different norms of initial data.In particular,the result mentioned in Theorem 1.5 offers an optimal decay rate of solutions,if the initial data belongs to the assigned weighted Sobolev space.In the proof of the theorem we reduce the estimate of solutions of a hyperbolic system to the corresponding case for a scalar pseudodifferential equation of the first order,and then establish the required estimate by using microlocal analysis.展开更多
This article is devoted to the study of variable 2-microlocal Besov-type and Triebel- Lizorkin-type spaces. These variable function spaces are defined via a Fourier-analytical approach. The authors then characterize t...This article is devoted to the study of variable 2-microlocal Besov-type and Triebel- Lizorkin-type spaces. These variable function spaces are defined via a Fourier-analytical approach. The authors then characterize these spaces by means of Q-transforms, Peetre maximal functions, smooth atoms, ball means of differences and approximations by analytic functions. As applications, some re- lated Sobolev-type embeddings and trace theorems of these spaces are Mso established. Moreover, some obtained results, such as characterizations via approximations by analytic functions, are new even for the classical variable Besov and Triebel-Lizorkin spaces.展开更多
基金Project supported by the Tianyuan Foundation of China and the National Foundation of China.
文摘Microlocal defect measures have been used to study the energy densities of scmilinear wave equations with oscillatory initial data.An "open problem" in literature is solved.
基金the National Natural Science Foundation of China.
文摘A commutative Noetherian ring R is called a regular Noetherian ring with pure dimension n, if for any maximal ideal m of R, gl.dimR_m=n, where R_m is the localization of R at the maximal ideal m. It is well known that if R is a finitely generated commutative algebra over some field, R is integral and gl. dimR【∞, then R is a regular Noetherian ring with pure dimension. Let D(V) be the ring of differential operators over the non-singular n-dimensional irreducible algebraic variety V. Then gr(D(V)) is a
文摘In this paper, we develop semi-classical analysis on H-type groups. We define semi-classical pseudodifferential operators, prove the boundedness of their action on square integrable functions and develop a symbolic calculus. Then, we define the semi-classical measures of bounded families of square integrable functions which consist of a pair formed by a measure defined on the product of the group and its unitary dual, and by a field of trace class positive operators acting on the Hilbert spaces of the representations. We illustrate the theory by analyzing examples, which show in particular that this semi-classical analysis takes into account the finite-dimensional representations of the group, even though they are negligible with respect to the Plancherel measure.
基金Project supported by the National Natural Science Foundation of China.
文摘In this paper, the author applies the Egorov Theorem for paradifferential operators established by the theory of para-Fourier integral operators to reduce microlocally paradifferential operators with the principal type. This reduction is used to study the propagation of singularities for solutions to nonlinear differential equations in the lower frequency range.
基金Research supported by grants of the Natural Science Foundation of Chinathe State Education Committee and the Huacheng Foundation.
文摘In this note, we use the so-called microlocal energy method to give a characterization of the Gevrey-Sobolev wave front set WF<sub>(</sub>H<sub>T,σ</sub><sup>S</sup> （u）, which will be useful in the study of non-linear microlocal analysis in Gevrey classes.
基金This work is partly supported by NNSF of China Doctoral Programme Foundation of IHEC
文摘In this paper we generalize the global Sobolev inequality introduced by Klainerman in studying wave equation to the hyperbolic system case.We obtain several decay estimates of solutions of a hyperbolic system of first order by different norms of initial data.In particular,the result mentioned in Theorem 1.5 offers an optimal decay rate of solutions,if the initial data belongs to the assigned weighted Sobolev space.In the proof of the theorem we reduce the estimate of solutions of a hyperbolic system to the corresponding case for a scalar pseudodifferential equation of the first order,and then establish the required estimate by using microlocal analysis.
基金Supported by the National Natural Science Foundation of China(Grant Nos.11571039,11671185,11471042 and 11701174)supported by the Construct Program of the Key Discipline in Hu’nan Province+1 种基金the Scientific Research Fund of Hu’nan Provincial Education Department(Grant No.17B159)the Scientific Research Foundation for Ph.D.Hu’nan Normal University(Grant No.531120-3257)
文摘This article is devoted to the study of variable 2-microlocal Besov-type and Triebel- Lizorkin-type spaces. These variable function spaces are defined via a Fourier-analytical approach. The authors then characterize these spaces by means of Q-transforms, Peetre maximal functions, smooth atoms, ball means of differences and approximations by analytic functions. As applications, some re- lated Sobolev-type embeddings and trace theorems of these spaces are Mso established. Moreover, some obtained results, such as characterizations via approximations by analytic functions, are new even for the classical variable Besov and Triebel-Lizorkin spaces.
