Noncommutative harmonic analysis on semigroup and ultracontractivity

TL;DR

This paper extends classical harmonic analysis results to noncommutative spaces, establishing equivalences between ultracontractivity of semigroups and Sobolev embedding properties of their generators, with applications to spectral multipliers and inequalities.

Contribution

It introduces noncommutative analogues of ultracontractivity and Sobolev embeddings, linking semigroup properties to spectral multiplier theorems and inequalities.

Findings

Characterization of $ ext{phi}$-ultracontractivity via Sobolev embeddings.

Spectral multiplier theorems for noncommutative generators.

Equivalence between ultracontractivity and logarithmic Sobolev inequalities.

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 $\phi$ 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.