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### Open Access Journals

DCDS

We establish an improved form of the classical logarithmic Sobolev inequality for the Gaussian
measure restricted to probability densities which satisfy a Poincaré inequality.
The result implies a lower bound
on the deficit in terms of the quadratic Kantorovich-Wasserstein distance. We similarly investigate
the deficit in the Talagrand quadratic transportation cost inequality this time by means of an
${ L}^1$-Kantorovich-Wasserstein distance, optimal for product measures, and deduce a lower bound on the deficit in the logarithmic Sobolev inequality in terms of this metric. Applications are given in the context
of the Bakry-Émery theory and the coherent state transform. The proofs combine tools from
semigroup and heat kernel theory and optimal mass transportation.

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