Nelson's gaussian hypercontractivity inequality captures the regularising property of the Ornstein--Uhlenbeck semigroup. Although early motivation arose from quantum field theory, the hypercontractivity inequality is wide-reaching and enjoys connections to a variety of topics. In this talk, we discuss a refinement of the hypercontractivity inequality when the class of input functions is restricted. Making contact with recent work of Eldan, Lehec and Shenfeld, we will derive certain deficit estimates for the log-Sobolev inequality from our refined version of hypercontractivity. This talk is based on joint work with Shohei Nakamura and Hiroshi Tsuji (both at Osaka).

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