XVI: 09, 138-150, LNM 920 (1982)
BAKRY, Dominique;
MEYER, Paul-André
Sur les inégalités de Sobolev logarithmiques (two parts) (
Applications of martingale theory)
These two papers are variations on a paper of G.F. Feissner (
Trans. Amer Math. Soc.,
210, 1965). Let $\mu$ be a Gaussian measure, $P_t$ be the corresponding Ornstein-Uhlenbeck semigroup. Nelson's hypercontractivity theorem states (roughly) that $P_t$ is bounded from $L^p(\mu)$ to some $L^q(\mu)$ with $q\ge p$. In another celebrated paper, Gross showed this to be equivalent to a logarithmic Sobolev inequality, meaning that if a function $f$ is in $L^2$ as well as $Af$, where $A$ is the Ornstein-Uhlenbeck generator, then $f$ belongs to the Orlicz space $L^2Log_+L$. The starting point of Feissner was to translate this again as a result on the ``Riesz potentials'' of the semi-group (defined whenever $f\in L^2$ has integral $0$) $$R^{\alpha}={1\over \Gamma(\alpha)}\int_0^\infty t^{\alpha-1}P_t\,dt\;.$$ Note that $R^{\alpha}R^{\beta}=R^{\alpha+\beta}$. Then the theorem of Gross implies that $R^{1/2}$ is bounded from $L^2$ to $L^2Log_+L$. This suggests the following question: which are in general the smoothing properties of $R^\alpha$? (Feissner in fact considers a slightly different family of potentials).\par The complete result then is the following : for $\alpha$ complex, with real part $\ge0$, $R^\alpha$ is bounded from $L^pLog^r_+L$ to $L^pLog^{r+p\alpha}_+L$. The method uses complex interpolation between two cases: a generalization to Orlicz spaces of a result of Stein, when $\alpha$ is purely imaginary, and the case already known where $\alpha$ has real part $1/2$. The first of these two results, proved by martingale theory, is of a quite general nature
Keywords: Logarithmic Sobolev inequalities,
Hypercontractivity,
Gaussian measures,
Riesz potentialsNature: Original
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XXI: 09, 173-175, LNM 1247 (1987)
ÉMERY, Michel;
YUKICH, Joseph E.
A simple proof of the logarithmic Sobolev inequality on the circle (
Real analysis)
The same kind of semi-group argument as in Bakry-Émery
1912 gives an elementary proof of the logarithmic Sobolev inequality on the circle
Keywords: Logarithmic Sobolev inequalitiesNature: New proof of known results Retrieve article from Numdam
