Ultracontractivity and strong Sobolev inequality for q-Ornstein-Uhlenbeck semigroup (-1 < q < 1)

We prove that q-Ornstein-Uhlenbeck semigroup U-t((q)) = Gamma(q)(e(-t) I) maps L-2 into L-infinity and parallel to U(t)((q))f parallel to infinity less than or equal to C(q)t(-3/2)parallel to f parallel to(2), 0 < t < 1 for -1 < q < 1. This is an extension of Ph. Biane's(3) result on Nelson hypercontractivity and the ultracontractivity of the free (q = 0) and q-second quantization.