Logarithmic-Sobolev and multilinear Hӧlder’s inequalities via heat flow monotonicity formulas

A, Abolarinwa and N.K, Oladejo and S.O, Salawu and C.A, Onate Logarithmic-Sobolev and multilinear Hӧlder’s inequalities via heat flow monotonicity formulas. Applied Mathematics and Computation, 364. p. 124640. ISSN 0096-3003

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Abstract

Heat flow monotonicity formulas have evolved in recent years as a powerful tool in deriv- ing 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 monotonic- ity formula is intimately related to the concavity of the power of Shannon entropy and Fisher Information, from which the associated logarithmic Sobolev inequality for probabil- ity measure in Euclidean setting is recovered. The second monotonicity formula combines very well with convolution and diffusion semigroup properties of the heat kernel to estab- lish the proof of the multilinear Hölder inequalities

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