On logarithmic Sobolev constant for diffusion semigroups (Q1091587)

This paper is concerned with estimates for the constant in the logarithmic Sobolev inequality. As an application, we show that for the Laguerre semigroup with the generator \(xD^ 2-(x-\lambda)D\), the logarithmic Sobolev constant \(\alpha_{\lambda}\) satisfies \(\log (3+6/\lambda)/\log 3\leq \alpha_{\lambda}16 \log (3+6/\lambda)\) for \(0<\lambda <\) and \(\alpha_{\lambda}=4\) for \(\leq \lambda <\infty\).