Abstract
In this paper, we prove some type of logarithmic Sobolev inequalities (with parameters) for operators in semi-direct product forms (see Sect. 1 for a precise definition). This generalizes the tensorization procedure for this type of inequalities and allows to deal with some operators with varying coefficients. We provide many examples of applications and obtain ultracontractive bounds for some of these operators by using appropriate Hardy’s type inequalities necessary for our method. This theory is developed in the setting of carre du champ with diffusion property.
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