Abstract
Heat flow monotonicity formulas have evolved in recent years as a powerful tool in deriving functional and geometric inequalities which are in turn useful in mathematical analysis and applications. This paper aims mainly at proving Logarithmic Sobolev and multilinear Hölder’s inequalities through the heat flow method. Precisely, two entropy monotonicity formulas are constructed via the heat flow. It is shown that the first entropy monotonicity formula is intimately related to the concavity of the power of Shannon entropy and Fisher Information, from which the associated logarithmic Sobolev inequality for probability measure in Euclidean setting is recovered. The second monotonicity formula combines very well with convolution and diffusion semigroup properties of the heat kernel to establish the proof of the multilinear Hölder inequalities.
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