In this paper we consider the divergence parabolic equation with bounded and measurable coefficients related to Hörmander's vector fields and establish a Nash type result,i.e.,the local Hölder regularity...In this paper we consider the divergence parabolic equation with bounded and measurable coefficients related to Hörmander's vector fields and establish a Nash type result,i.e.,the local Hölder regularity for weak solutions.After deriving the parabolic Sobolev inequality,(1,1)type Poincaré inequality of Hörmander's vector fields and a De Giorgi type Lemma,the Hölder regularity of weak solutions to the equation is proved based on the estimates of oscillations of solutions and the isomorphism between parabolic Campanato space and parabolic Hölder space.As a consequence,we give the Harnack inequality of weak solutions by showing an extension property of positivity for functions in the De Giorgi class.展开更多
The massive vector bosons Z o, W ± and the scalar Higgs-boson H o assumed in weak interaction theory, but also the six quarks required in strong interactions are well understood in an alternative quantum field th...The massive vector bosons Z o, W ± and the scalar Higgs-boson H o assumed in weak interaction theory, but also the six quarks required in strong interactions are well understood in an alternative quantum field theory of fermions and bosons: Z o and W ± as well as all quark-antiquark states (here only the tt¯state is discussed) are described by bound states with scalar coupling between their massless constituents and have a structure similar to leptons. However, the scalar Higgs-boson H o corresponds to a state with vector coupling between the elementary constituents. Similar scalar states are expected also in the mass region of the mesons ω (0.782 GeV) - Υ ( 9.46 GeV). The underlying calculations can be run on line using the Web-address https://h2909473.stratoserver.net.展开更多
In this paper,we obtain a parameter type logarithmic Sobolev inequality with weight and a modified parameter type logarithmic Sobolev inequality with weight on Euclidean space based on the parameter type logarithmic C...In this paper,we obtain a parameter type logarithmic Sobolev inequality with weight and a modified parameter type logarithmic Sobolev inequality with weight on Euclidean space based on the parameter type logarithmic Caffarelli-Kohn-Nirenberg inequality on Euclidean space,respectively.By virtue of the convexity of a special function equivalent to a logarithmic Hölder inequality,and combining the Sobolev inequality and Gagliardo-Nirenberg inequality on Hörmander's vector fields,we also derive a logarithmic Sobolev inequality and a parameter type logarithmic Gagliardo-Nirenberg inequality on Hörmander's vector fields,respectively.In addition,a parameter type logarithmic Sobolev inequality on Hörmander's vector fields is given by suitable stretching transformation.展开更多
基金This work is supported by the National Natural Science Foundation of China(No.1177-1354).
文摘In this paper we consider the divergence parabolic equation with bounded and measurable coefficients related to Hörmander's vector fields and establish a Nash type result,i.e.,the local Hölder regularity for weak solutions.After deriving the parabolic Sobolev inequality,(1,1)type Poincaré inequality of Hörmander's vector fields and a De Giorgi type Lemma,the Hölder regularity of weak solutions to the equation is proved based on the estimates of oscillations of solutions and the isomorphism between parabolic Campanato space and parabolic Hölder space.As a consequence,we give the Harnack inequality of weak solutions by showing an extension property of positivity for functions in the De Giorgi class.
文摘The massive vector bosons Z o, W ± and the scalar Higgs-boson H o assumed in weak interaction theory, but also the six quarks required in strong interactions are well understood in an alternative quantum field theory of fermions and bosons: Z o and W ± as well as all quark-antiquark states (here only the tt¯state is discussed) are described by bound states with scalar coupling between their massless constituents and have a structure similar to leptons. However, the scalar Higgs-boson H o corresponds to a state with vector coupling between the elementary constituents. Similar scalar states are expected also in the mass region of the mesons ω (0.782 GeV) - Υ ( 9.46 GeV). The underlying calculations can be run on line using the Web-address https://h2909473.stratoserver.net.
基金National Natural Science Foundation of China(Grant No.12261053)the project Science and Technology Project of Yunnan Province,Key Technology Projects in Yunnan Province(Grant No.202302AF080003)+1 种基金the Special Basic Cooperative Research Programs of Yunnan Provincial Undergraduate Universities Association(Grant Nos.202401BA070001-110,202301BA070001-002 and 202101BA070001-132)the Scientific Research Fund of Education Department of Yunnan Province(Grant Nos.2024Y775,2024Y776,2025Y1076 and 2024J0775).
文摘In this paper,we obtain a parameter type logarithmic Sobolev inequality with weight and a modified parameter type logarithmic Sobolev inequality with weight on Euclidean space based on the parameter type logarithmic Caffarelli-Kohn-Nirenberg inequality on Euclidean space,respectively.By virtue of the convexity of a special function equivalent to a logarithmic Hölder inequality,and combining the Sobolev inequality and Gagliardo-Nirenberg inequality on Hörmander's vector fields,we also derive a logarithmic Sobolev inequality and a parameter type logarithmic Gagliardo-Nirenberg inequality on Hörmander's vector fields,respectively.In addition,a parameter type logarithmic Sobolev inequality on Hörmander's vector fields is given by suitable stretching transformation.
