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Several logarithmic Caffarelli-Kohn-Nirenberg inequalities and applications. (English) Zbl 1373.35014

Summary: 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.

MSC:

35A23 Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals
35R03 PDEs on Heisenberg groups, Lie groups, Carnot groups, etc.
26D10 Inequalities involving derivatives and differential and integral operators
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