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Building on the inequalities for homogeneous tetrahedral polynomials inindependent Gaussian variables due to R. Lata{\l}a we provide a concentrationinequality for non-necessarily Lipschitz functions $f\colon \R^n \to \R$ withbounded derivatives of higher orders, which hold when the underlying measuresatisfies a family of Sobolev type inequalities $\|g- \E g\|_p \le C(p)\|\nablag\|_p.$  Such Sobolev type inequalities hold, e.g., if the underlying measuresatisfies the log-Sobolev inequality (in which case $C(p) \le C\sqrt{p}$) orthe Poincar\'e inequality (then $C(p) \le Cp$). Our concentration estimates areexpressed in terms of tensor-product norms of the derivatives of $f$.  When the underlying measure is Gaussian and $f$ is a polynomial(non-necessarily tetrahedral or homogeneous), our estimates can be reversed (upto a constant depending only on the degree of the polynomial). We also showthat for polynomial functions, analogous estimates hold for arbitrary randomvectors with independent sub-Gaussian coordinates.  We apply our inequalities to general additive functionals of random vectors(in particular linear eigenvalue statistics of random matrices) and the problemof counting cycles of fixed length in Erd\H{o}s-R{\'e}nyi random graphs,obtaining new estimates, optimal in a certain range of parameters.

Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order