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Heat equation and convolution inequalities. (English) Zbl 1303.26023

Summary: It is known that many classical inequalities linked to convolutions can be obtained by looking at the monotonicity in time of convolutions of powers of solutions to the heat equation, provided that both the exponents and the coefficients of diffusions are suitably chosen and related. This idea can be applied to give an alternative proof of the sharp form of the classical Young’s inequality and its converse, to Brascamp-Lieb type inequalities, Babenko’s inequality and Prékopa-Leindler inequality as well as the Shannon’s entropy power inequality. This note aims in presenting new proofs of these results, in the spirit of the original arguments introduced by A. J. Stam [Inform. and Control 2, 101–112 (1959; Zbl 0085.34701)] to prove the entropy power inequality.

MSC:

26D15 Inequalities for sums, series and integrals
39B62 Functional inequalities, including subadditivity, convexity, etc.
94A17 Measures of information, entropy

Citations:

Zbl 0085.34701
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References:

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