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 μ-deformed Segal-Bargmann space and arises in the works of Wigner, Rosenblum, and Marron. To achieve this we define μ-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 Lp bounded, and we conjecture that it is Lp contractive and, even more strongly, that it is hypercontractive.