Noncommutative harmonic analysis on semigroup and ultracontractivity

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 $\phi$ and $\psi$ be regularly related. We prove that the semigroup $(T_t)_{t>0}$ is $\phi$-ultracontractive, i.e. $\|T_t x\|_\infty \leq C \phi(t)^{-1} \|x\|_1$ for all $x\in L_1(\mathcal{M})$ and $t>0$ if and only if its infinitesimal generator $L$ has the Sobolev embedding properties: $\|\psi(L)^{-\alpha} x\|_q \leq C'\|x\|_p$ 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 $\phi$-ultracontractive semigroup. We also show the equivalence between $\phi$-ultracontractivity and logarithmic Sobolev inequality for some special $\phi$. Finally, we gives some results on local ultracontractivity.