Caltech/UCLA Joint Analysis Seminar

UCLA MS 6627

The necessary and sufficient conditions will be obtained on L p Lp L^p to L q Lq L^q boundedness of the Hermite semigroup e zΔ ezΔ e^{z\Delta} on the boolean cube of an arbitrary dimension equipped with uniform counting measure, where z z z is a complex number and 1≤p≤q 1≤p≤q 1\leq p \leq q . This solves an old open problem in complex hypercontractivity theory on the Hamming. Certain cases of the triples (p,q,z) (p,q,z) (p,q,z) were characterized by Bonami (1970); Beckner (1975); and Weissler (1979). Several applications will be presented. Work in progress with Fedja Nazarov.