Logarithmic Sobolev inequalities for the heat-diffusion semigroup

Abstract:

An explicit formula relating the Hermite semigroup ${e^{ - tH}}$ on R with Gauss measure and the heat-diffusion semigroup ${e^{t\Delta }}$ on R with Lebesgue measure is proved. From this formula it follows that Nelson’s hypercontractive estimates for ${e^{ - tH}}$ are equivalent to the best norm estimates for ${e^{t\Delta }}$ as a map ${L^q}(R)$ into ${L^p}(R),1 < q < p < \infty$. Furthermore,the inequality \[ \frac {d}{{dq}}\log \left \| \phi \right \|_q^q \leqslant \frac {n}{{2q}}\log \left [ {\frac {{{q^2}}}{{2\pi ne(q - 1)}} \cdot \frac {{\operatorname {Re} \langle - \Delta \phi ,{J^q}\phi \rangle }}{{\left \| \phi \right \|_q^q}}} \right ] + \log {\left \| \phi \right \|_q},\] where $1 < q < \infty ,{J^q}\phi = (\operatorname {sgn} \phi )|\phi {|^{q - 1}}$, and the norms and sesquilinear form $\langle ,\rangle$ are taken with respect to Lebesgue measure on ${R^n}$, is shown to be equivalent to the best norm estimates for ${e^{t\Delta }}$ as a map from ${L^q}({R^n})$ into ${L^p}({R^n})$. This inequality is analogous to Gross’ logarithmic Sobolev inequality. Also, the above inequality is compared with a classical Sobolev inequality.