Noncommutative Harmonic Analysis on Semigroup and Ultracontractivity

We extend some classical results of Cowling and Meda to the noncommutative setting. Let (Tt)t>0 be a symmetric contraction semigroup on a noncommutative space Lp(M), and let the functions φ and ψ be regularly related. We prove that the semigroup (Tt)t>0 is φ-ultracontractive, i.e. ‖Ttx‖∞ ≤ Cφ(t)‖x‖1 for all x ∈ L1(M) and t > 0 if and only if its infinitesimal generator L has the Sobolev embedding properties: ‖ψ(L)x‖q ≤ C′‖x‖p for all x ∈ Lp(M), where 1 < p < q < ∞ and α = 1 p − 1 q . We establish some noncommutative spectral multiplier theorems and maximal function estimates for generator of φ-ultracontractive semigroup. We also show the equivalence between φ-ultracontractivity and logarithmic Sobolev inequality for some special φ. Finally, we gives some results on local ultracontractivity.