We present precise multilevel exponential concentration inequalities for polynomials in Ising models satisfying the Dobrushin condition. The estimates have the same form as two-sided tail estimates for polynomials in Gaussian variables due to Latała. In particular, for quadratic forms we obtain a Hanson-Wright type inequality.We also prove concentration results for convex functions and estimates for nonnegative definite quadratic forms, analogous as for quadratic forms in i.i.d. Rademacher variables, for more general random vectors satisfying the approximate tensorization property for entropy.2010 Mathematics Subject Classification. 60E15, 82B99.

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We give a sufficient and necessary condition for a probability measure µ on the real line to satisfy the logarithmic Sobolev inequality for convex functions. The condition is expressed in terms of the unique left-continuous and non-decreasing map transporting the symmetric exponential measure onto µ. The main tool in the proof is the theory of weak transport costs.As a consequence, we obtain dimension-free concentration bounds for the lower and upper tails of convex functions of independent random variables which satisfy the convex log-Sobolev inequality.Due to its tensorization property the log-Sobolev inequality is a powerful tool and can be used to obtain dimension-free concentration bounds (via the Herbst argument). It has been investigated also in more general settings of Riemannian manifolds and in context of applications to the study of Markov chains, see e.g. the monographs [3,4] and the expository article [11].

Abstract. We calculate the norms of the operators connected to the action of the BeurlingAhlfors transform on radial function subspaces introduced by Bañuelos and Janakiraman. In particular, we find the norm of the Beurling-Ahlfors transform acting on radial functions for p > 2, extending the results obtained by Bañuelos and Janakiraman, Bañuelos and Osȩkowski, and Volberg for 1 < p ≤ 2.

We prove that for a probability measure on R n , the Poincaré inequality for convex functions is equivalent to the weak transportation inequality with a quadratic-linear cost. This generalizes recent results by Gozlan, Roberto, Samson, Shu, Tetali and Feldheim, Marsiglietti, Nayar, Wang, concerning probability measures on the real line.The proof relies on modified logarithmic Sobolev inequalities of Bobkov-Ledoux type for convex and concave functions, which are of independent interest.We also present refined concentration inequalities for general (not necessarily Lipschitz) convex functions, complementing recent results by Bobkov, Nayar, and Tetali.

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