Modified log-Sobolev inequalities for convex functions on the real line. Sufficient conditions

A characterization of a class of convex log-Sobolev inequalities on the real line

“…This condition is expressed in terms of the unique left-continuous and non-decreasing map transporting the symmetric exponential measure onto µ (more precisely, in terms of the quantity ∆ µ (h) defined below in (1.4); see Section 4 for a precise statement of the result). In fact, their sufficient condition is weaker than the condition considered in [2] (and leads to a result formally stronger than the convex log-Sobolev inequality, cf. Proposition 4.1).…”

“…In [1] Adamczak found a sufficient condition for a probability measure on the real line to satisfy the convex log-Sobolev inequality with H(x) = x 2 , x ∈ R. This has been extended to functions of the form H(x) = max{x 2 , x (β+1)/β }, where β ∈ [0, 1], by Adamczak and the second named author [2].…”

“…In this section we establish the equivalence of the convex log-Sobolev inequality and the weak transport-entropy inequality. In the case of the quadratic cost this was done in [16] (see also [2] for related results for other cost functions). Using the techniques developed therein, in especially the dual formulation (2.2), we extend this result to a wider class of cost functions.…”

“…We work with measures on the real line, but in contrast to Section 4 there are no essential problems with extending the results of this section to a higher dimensional setting (cf. [16,2]).…”

Concentration of Measure Principle and Entropy-Inequalities

“…As for the usual transport costs, connections have been established between barycentric transport inequalities and logarithmic Sobolev inequalities restricted to a class of functions (see [GRST14b,AS15]). To simplify, in this section we only consider the case θ(h) = h 2 where · is a fixed norm on R n .…”