Free hypercontractivity

We prove the analog of Nelson's optimal hypercontractivity inequality, for the free second quantization (in the sense of Voiculescu) of a contraction between two real Hilbert spaces. The proof extends also to the case of the q-quantization functors of Bozejko, Kummerer and Speicher, for q is an element of [-1, 1], interpolating between the bosonic and the fermionic functors which correspond to q = + 1 and q = -1 respectively.