Reverse inequalities in μ-deformed Segal-Bargmann analysis

We prove reverse hypercontractivity inequalities as well as reverse log-Sobolev inequalities in the context of a space of holomorphic functions, which is called the mu-deformed Segal-Bargmann space and arises in the works of Wigner, Rosenblum, and Marron. To achieve this we define mu-deformations of energy and entropy. Our principle results generalize earlier works of Carlen and Sontz. We also show that the semigroup of this theory is L-p bounded, and we conjecture that it is L-p contractive and, even more strongly, that it is hypercontractive. (c) 2006 American Institute of Physics.