An example in the theory of hypercontractive semigroups

Let $L = x({d^2}/d{x^2}) + (1 - x)(d/dx)$ on ${C_c}((0,\infty ))$ be the Laguerre operator. It is shown that for $t > 0$, and $1 < p < q < \infty ,\;{e^{tl}}:{L^p}({e^{ - x}}dx) \to {L^q}({e^{ - x}}dx)$ has norm 1 if and only if ${e^{ - t}} \leqslant (p - 1)/(q - 1)$ and the corresponding logarithmic Sobolev constant is not equal to $2/\lambda$, where $\lambda$ is the smallest nonzero eigenvalue of $L$.