Several logarithmic Caffarelli-Kohn-Nirenberg inequalities and applications

In this paper, based on the Caffarelli-Kohn-Nirenberg inequalities on the Euclidean space and the weighted Hölder inequality, we establish the logarithmic Caffarelli-Kohn-Nirenberg inequalities and parameter type logarithmic Caffarelli-Kohn-Nirenberg inequalities, and give applications for the weighted ultracontractivity of positive strong solutions to a kind of evolution equations. We also prove corresponding logarithmic Caffarelli-Kohn-Nirenberg inequalities and parameter type logarithmic Caffarelli-Kohn-Nirenberg inequalities on the Heisenberg group and related to generalized Baouendi-Grushin vector fields. Some applications are provided.