  In this paper we prove multilevel concentration inequalities for bounded
functionals $f = f(X_1, \ldots, X_n)$ of random variables $X_1, \ldots, X_n$
that are either independent or satisfy certain logarithmic Sobolev
inequalities. The constants in the tail estimates depend on the operator norms
of $k$-tensors of higher order differences of $f$.
  We provide applications in both dependent and independent random variables.
This includes deviation inequalities for empirical processes $f(X) = \sup_{g
\in \mathcal{F}} \lvert g(X) \rvert$ and suprema of homogeneous chaos in
bounded random variables in the Banach space case given by $f(X) = \sup_{t}
\lVert \sum_{i_1 \neq \ldots \neq i_d} t_{i_1 \ldots i_d} X_{i_1} \cdots
X_{i_d}\rVert_{\mathcal{B}}$. The latter application is comparable to earlier
results of Boucheron-Bousquet-Lugosi-Massart and provides the upper tail bounds
of Talagrand. In the case of Rademacher random variables, we give an
interpretation of the results in terms of quantities familiar in Boolean
analysis. Further applications are concentration inequalities for
$U$-statistics with bounded kernels $h$ and for the number of triangles in an
exponential random graph model.
