  We study a class of logarithmic Sobolev inequalities with a general form of
the energy functional. The class generalizes various examples of modified
logarithmic Sobolev inequalities considered previously in the literature.
Refining a method of Aida and Stroock for the classical logarithmic Sobolev
inequality, we prove that if a measure on $\mathbb{R}^n$ satisfies a modified
logarithmic Sobolev inequality then it satisfies a family of $L^p$-Sobolev-type
inequalities with non-Euclidean norms of gradients (and dimension-independent
constants). The latter are shown to yield various concentration-type estimates
for deviations of smooth (not necessarily Lipschitz) functions and measures of
enlargements of sets corresponding to non-Euclidean norms. We also prove a
two-level concentration result for functions of bounded Hessian and measures
satisfying the classical logarithmic Sobolev inequality.

Probability;Functional Analysis