Noncommutative harmonic analysis on semigroup and ultracontractivity

Original · EN

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. Tₜ x∞ ≤ C ϕ(t)⁻¹ x₁ for all x∈ L₁(M) and t>0 if and only if its infinitesimal generator L has the Sobolev embedding properties: ψ(L)⁻α xq ≤ C'xₚ 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.