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 $\varphi$ and $\psi$ be regularly related. We prove that the semigroup $(T_t{)}_{t>0}$ is $\varphi$-ultracontractive, i.e. $\mathrm{\parallel }{T}_{t}x{\mathrm{\parallel }}_{\mathrm{\infty }}\le C\varphi (t{)}^{-1}\mathrm{\parallel }x{\mathrm{\parallel }}_1$ for all $x\in L_1(\mathcal{M})$ and $t>0$ if and only if its infinitesimal generator $L$ has the Sobolev embedding properties: $|\psi(L)^{-\alpha} x|_q \leq C’|x|_p$forall$x\in L_p($\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.