Hypercontractivity and its usage in semigroup theory.
(L’hypercontractivité et son utilisation en théorie des semigroupes.)

*(French)*Zbl 0856.47026
Bakry, Dominique (ed.) et al., Lectures on probability theory. Ecole d’Eté de Probabilités de Saint-Flour XXII-1992. Summer School, 9th-25th July, 1992, Saint-Flour, France. Berlin: Springer. Lect. Notes Math. 1581, 1-114 (1994).

It is the aim of the paper to present a survey of results and methods around the topics of sub-Markovian semigroups, hypercontractivity, and logarithmic Sobolev inequalities. The paper reflects the author’s preferences lying in the area of probability. This fact makes the treatment an interesting and valuable supplement to E. B. Davies’ book “Heat kernels and spectral theory” (1989; Zbl 0699.35006). In view of the richness of the material covered in this article it seems to be adequate to sketch the topics of the paper by giving an extended table of contents:

1. Ornstein-Uhlenbeck semigroup, relations to the Fourier transformation, approximation by the coin tossing process, Nelson’s hypercontractivity theory, Beckner’s proof of Babenko’s inequality (best constant in the Hausdorff-Young inequality).

2. Positivity preserving semigroups, Markov semigroups, symmetric semigroups, core theorems for the generator, the carré-du-champ operator, diffusion semigroups, invariance and reversibility, spectral gap.

3. \(p\)-logarithmic Sobolev inequalities and hypercontractivity (Gross’ method), tense (i.e., ‘tendues’) \(p\)-logarithmic Sobolev inequalities and the spectral gap, hypercontractivity in the complex domain.

4. Sobolev inequalities, relation to logarithmic Sobolev inequalities, uniform estimates for heat kernels (method of Davies and Simon), behaviour of the semigroup at zero Sobolev inequalities (results of Varopoulos), weak Sobolev inequalities, lower and upper bounds for heat kernels.

5. Non-uniform estimates for heat kernels in terms of the intrinsic distance (Davies’ method), Sobolev inequalities and growth of volumes of balls.

6. Intrinsic curvature and dimension of diffusion, the operator \(\Gamma_2\), calculation of these quantities for elliptic diffusions on manifolds, connection with (logarithmic, weak) Sobolev inequalities.

For the entire collection see [Zbl 0797.00021].

1. Ornstein-Uhlenbeck semigroup, relations to the Fourier transformation, approximation by the coin tossing process, Nelson’s hypercontractivity theory, Beckner’s proof of Babenko’s inequality (best constant in the Hausdorff-Young inequality).

2. Positivity preserving semigroups, Markov semigroups, symmetric semigroups, core theorems for the generator, the carré-du-champ operator, diffusion semigroups, invariance and reversibility, spectral gap.

3. \(p\)-logarithmic Sobolev inequalities and hypercontractivity (Gross’ method), tense (i.e., ‘tendues’) \(p\)-logarithmic Sobolev inequalities and the spectral gap, hypercontractivity in the complex domain.

4. Sobolev inequalities, relation to logarithmic Sobolev inequalities, uniform estimates for heat kernels (method of Davies and Simon), behaviour of the semigroup at zero Sobolev inequalities (results of Varopoulos), weak Sobolev inequalities, lower and upper bounds for heat kernels.

5. Non-uniform estimates for heat kernels in terms of the intrinsic distance (Davies’ method), Sobolev inequalities and growth of volumes of balls.

6. Intrinsic curvature and dimension of diffusion, the operator \(\Gamma_2\), calculation of these quantities for elliptic diffusions on manifolds, connection with (logarithmic, weak) Sobolev inequalities.

For the entire collection see [Zbl 0797.00021].

Reviewer: J.Voigt (Dresden)

##### MSC:

47D07 | Markov semigroups and applications to diffusion processes |

47D06 | One-parameter semigroups and linear evolution equations |

35K15 | Initial value problems for second-order parabolic equations |