Ultracontractive bounds on Hamilton-Jacobi solutions (Q1865007)

The main results of this paper concern the border of three theorems, which are respectively the theorem of hypercontractivity of Gross, the theorem of hypercontractivity of Bobkov-Gentil-Ledoux and the theorem of ultracontractivity of Varopoulos. Following the equivalence between logarithmic Sobolev inequality, hypercontractivity of the heat semigroup showed by Gross and hypercontractivity of Hamilton-Jacobi equations, the author proves, like the Varopoulos theorem, the equivalence between Euclidean-type Sobolev inequality and an ultracontractive control of the Hamilton-Jacobi equations. Ultracontractive estimations under general Sobolev inequality, which imply in the particular case of probability measure transportation inequalities, are also obtained.