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 etΔ 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 tΔ as a map Lq(R) into $L^p(R), 1 < q < p < |infty$ . Furthermore, the inequality $\frac{d}{dq} \log\| \phi \|^q_q \leqslant \frac{n}{2q} \log\bigg\lbrack\frac {q^2} {2\pi ne(q - 1)} \cdot \frac {\operatorname{Re}\langle -\Delta\phi, J^q\phi\rangle} {\| \phi \|^q_q}\bigg\rbrack + \log\| \phi \|_q$ , where $1 < q < \infty, J^q\phi = (\operatorname{sgn} \phi)|\phi|^{q-1}$ , and the norms and sesquilinear form $\langle,\langle$ are taken with respect to Lebesgue measure on Rn, is shown to be equivalent to the best norm estimates for etΔ as a map from Lq(Rn) into Lp(Rn). This inequality is analogous to Gross'logarithmic Sobolev inequality. Also, the above inequality is compared with a classical Sobolev inequality.