Noncommutative harmonic analysis on semigroup and ultracontractivity

Abstract

We extend some classical results of Cowling and Meda to the noncommutative setting. Let $(T_t{)}_{t>0}$ be a symmetric contraction semigroup on a noncommutative space $L_p(\mathcal{M}),$ and let the functions $\varphi$ and $\psi$ be regularly related. We prove that the semigroup $(T_t{)}_{t>0}$ is $\varphi$-ultracontractive, i.e. $\Vert T_{t}x{\Vert }_{\infty }\le C\varphi (t{)}^{-1}\Vert x{\Vert }_1$ for all $x\in L_1(\mathcal{M})$ and $t>0$ if and only if its infinitesimal generator $L$ has the Sobolev embedding properties: $\Vert \psi (L{)}^{-\alpha }x{\Vert }_q\le C^{\prime }\Vert x{\Vert }_p$ for all $x\in L_p(\mathcal{M}),$ where $1<p<q<\infty$ and $\alpha =\frac{1}{p}-\frac{1}{q}.$ We establish some noncommutative spectral multiplier theorems and maximal function estimates for generator of $\varphi$-ultracontractive semigroup. We also show the equivalence between $\varphi$-ultracontractivity and logarithmic Sobolev inequality for some special $\varphi$. Finally, we gives some results on local ultracontractivity.