Noncommutative Logarithmic Sobolev Inequalities

We show that the logarithmic Sobolev inequality holds for an arbitrary hypercontractive semigroup \(\{e^{-tP}\}_{t\ge 0}\) ${\{e^{-tP}\}}_{t\ge 0}$ acting on a noncommutative probability space \(({\mathcal {M}},\tau )\) $(\mathcal{M},\tau )$ : \(\begin{aligned} \Vert x\Vert _{L_p(\log L)^{ps}({\mathcal {M}})}\le c_{p,s}\Vert P^s(x)\Vert _{L_p({\mathcal {M}})},\quad 1<p<\infty , \end{aligned}\) $$\begin{array}{c}\hfill {\Vert x\Vert }_{L_p{(logL)}^{\mathit{ps}}(\mathcal{M})}\le c_{p,s}{\Vert P^s(x)\Vert }_{L_p(\mathcal{M})},1<p<\infty ,\end{array}$$ for every mean zero x and \(0<s<\infty \) $0<s<\infty$ . By selecting \(s=1/2\) $s=1/2$ , one can recover the p-logarithmic Sobolev inequality whenever the Riesz transform is bounded. Our inequality applies to numerous concrete cases, including Poisson semigroups for free groups, the Ornstein-Uhlenbeck semigroup for mixed Q-gaussian von Neumann algebras, the free product for Ornstein-Uhlenbeck semigroups etc. This provides a unified approach for functional analysis form of logarithmic Sobolev inequalities in general noncommutative setting.