AN EXAMPLE OF IMPROVED HYPERCONTRACTIVITY.

The purpose of this presentation is to provide an example where improved in-formation on hypercontractivity can be achieved. Improved in the sense that the result emanating from a logarithmic Sobolev inequality is bootstrapped to get a better degree of hypercontractivity. The first section is devoted to the construction of a class of probability spaces together with operator semigroups that display a spectral gap between their only non-trivial eigenvalue and the corresponding logarithmic Sobolev exponent. The construction generalizes a three-point space used by the author in [A]. With a different interpretation similar probability spaces were built by Diaconis and Saloff-Coste in [DS]. All necessary background on hypercontractivity and logarithmic Sobolev inequalities are conveniently found in the exposition [G2]. The second section introduces two variants of Gross ’ theorem on how hypercon-tractivity may be deduced from logarithmic Sobolev inequalities. These variants will be be combined in section three in order to derive stronger hypercontractivity claims in any situation where a spectral gap is present, in particular to the mode