The author generalizes the Poincaré inequality \(\int | f|^ 2d\mu -(\int fd\mu)^ 2\leq \int | \nabla f|^ 2d\mu\) where \(\mu =\Pi (2\pi)^{-1/2}e^{-x^ 2_ k/2}dx_ k\) to the situation where \(N=-\Delta +x\cdot \nabla\) and, in general, we denote \(e^{tA}\) for the semigroup with generator A. His first result is Theorem 1. For \(f\in L^ 2(d\mu)\), \(1\leq p\leq 2\), and \(e^{-t}=\sqrt{p-1}\), \[ (1)\quad \int | f|^ 2d\mu -(\int e^{-tN}fd\mu)^ 2\leq (2-p)\int | \nabla f|^ 2d\mu \] and \[ (2)\quad \int | f|^ 2d\mu - (\int | f|^ pd\mu)^{2/p}\leq (2-p)\int | \nabla f|^ 2d\mu. \] Nelson’s hypercontractive estimate \(\int | e^{- tN}f|^ 2d\mu \leq (\| f\|_{L^ p(d\mu)})^ 2\) provides the link between the two inequalities. The Poincaré inequality is obtained at \(p=1\) and as p approaches 2, a limiting argument gives the logarithmic Sobolev estimate \[ \int | f|^ 2\ell n| f| d\mu -(\int | f|^ 2d\mu)\ell n(\int | f|^ 2d\mu)^{1/2}\leq \int | \nabla f|^ 2d\mu. \] The author gives a similar estimate for \(F\in L^ 2(S^ n)\) in terms of its harmonic extension u to the interior of the unit ball. The sharp estimate involves a convolution with the Poisson kernel in the analogue of (1) and (2) becomes \[ \int_{S^ n}| F|^ 2d\xi -(\int_{S^ n}| F|^ 2d\xi)^{2/p}\leq (2-p)/\omega_ n\int_{| x| \leq 1}| \nabla u|^ 2dx\leq (2-p)/n\int_{S^ n}| \nabla F|^ 2d\xi. \]