Several Parameter Type Logarithmic Sobolev Inequalities

Abstract

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 in-equality 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.