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Abstract

We establish functional type inequalities linking the regularity properties of sequences of operators $S=(S_n)$ acting on $L^2$-spaces with those of the canonical Gaussian process on the associated subsets of $L^2$ defined by $(S_n(f))$, $f\in L^2$. These inequalities allow us to easily deduce as corollaries Bourgain's famous entropy criteria in the theory of almost everywhere convergence. They also provide a better understanding of the role of the Gaussian processes in the study of almost everywhere convergence. A partial converse path to Bourgain's entropy criteria is also proposed.