Noncommutative Logarithmic Sobolev Inequalities

We show that the logarithmic Sobolev inequality holds for an arbitrary hypercontractive semigroup \(\{e^{-tP}\}_{t\ge 0}\) acting on a noncommutative probability space \(({\mathcal {M}},\tau )\):

$$\begin{aligned} \Vert x\Vert _{L_p(\log L)^{ps}({\mathcal {M}})}\le c_{p,s}\Vert P^s(x)\Vert _{L_p({\mathcal {M}})},\quad 1<p<\infty , \end{aligned}$$

for every mean zero x and \(0<s<\infty \). By selecting \(s=1/2\), 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.