2016/03/14 by Xiao Xiong, Xiong, Xiao
Mathematics · #46L53 #Advanced Harmonic Analysis Research #Advanced Operator Algebra Research #FOS: Mathematics #Operator Algebras (math.OA) #Spectral Theory in Mathematical Physics #math.OA #msc:46L53
paper · pdf · doi:10.48550/arxiv.1603.04247
openalex publication_date 2016/03/14 · arxiv created 2016/03/15 · arxiv updated 2016/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ‖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)-α 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.