On logarithmic Sobolev constant for diffusion semigroups

AbstractThis 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 xD2 − (x − λ) D, the logarithmic Sobolev constant αλ satisfies log(3 + 6λ)/log 3 ⩽ αλ ⩽ 16 log(3 + 6λ) for 0 < λ < 12and αλ = 4 for12 ⩽ λ < ∞