Related papers: Hypercontractivity and the logarithmic Sobolev ine…

We study the Sobolev inequality and the existence of its extremal functions in the setting of homogeneous H\"{o}rmander vector fields. A principal result establishes a mutual inclusion between the set of volume growth rates of subunit balls…

In a 2013 paper, the author showed that the convolution of a compactly supported measure on the real line with a Gaussian measure satisfies a logarithmic Sobolev inequality (LSI). In a 2014 paper, the author gave bounds for the optimal…

We prove stability results in hypercontractivity estimates for the Hopf--Lax semigroup in $\mathbb R^n$ and apply them to deduce stability results for the Euclidean $L^p$-logarithmic Sobolev inequality for any $p>1$. As a main tool, we use…

Using a non-negative curvature condition, we prove the complete version of modified log-Sobolev inequalities for central Markov semigroups on various compact quantum groups, including group von Neumann algebras, free orthogonal group and…

Consider a hyperbolic group G and a quasiconvex subgroup H of infinite index. We construct a set-theoretic section s of the quotient map (of sets) from G to G/H such that s(G/H) is a net in G; that is, any element of G is a bounded distance…

In this paper, we consider the cubic nonlinear Schrodinger equation, and the Hartree equation, with sufficiently regular convolution potential, both on the real line. We are interested in bounding the growth of high Sobolev norms of…

In this paper we show a number of logarithmic inequalities on several classes of Lie groups: log-Sobolev inequalities on general Lie groups, log-Sobolev (weighted and unweighted), log-Gagliardo-Nirenberg and log-Caffarelli-Kohn-Nirenberg…

We prove that the complete modified logarithmic Sobolev constant of a quantum Markov semigroup is bounded by the inverse of its complete positivity mixing time. For classical Markov semigroups, this implies that every sub-Laplacian given by…

An analogue of Gross' logarithmic Sobolev inequality for a class of elements of noncommutative two tori is proved.

We prove uniform boundedness of certain boundary representations on appropriate fractional Sobolev spaces $W^{s,p}$ with $p>1$ for arbitrary Gromov hyperbolic groups. These are closed subspaces of $L^p$ and in particular Hilbert spaces in…

The paper is devoted to provide Michael-Simon-type $L^p$-logarithmic-Sobolev inequalities on complete, not necessarily compact $n$-dimensional submanifolds $\Sigma$ of the Euclidean space $\mathbb R^{n+m}$. Our first result, stated for…

In the setting of Carnot groups, we prove the $q-$Logarithmic Sobolev inequality for probability measures as a function of the Carnot-Carath\'eodory distance. As an application, we use the Hamilton-Jacobi equation in the setting of Carnot…

We define a Hamilton-Jacobi semigroup acting on continuous functions on a compact length space. Following a strategy of Bobkov, Gentil and Ledoux, we use some basic properties of the semigroup to study geometric inequalities related to…

We consider a noninteracting unbounded spin system with conservation of the mean spin. We derive a uniform logarithmic Sobolev inequality (LSI) provided the single-site potential is a bounded perturbation of a strictly convex function. The…

We study limits at infinity for homogeneous Hajlasz-Sobolev functions defined on uniformly perfect metric spaces equipped with a doubling measure. We prove that a quasicontinuous representative of such a function has a pointwise limit at…

We introduce and study a notion of Orlicz hypercontractive semigroups. We analyze their relations with general $F$-Sobolev inequalities, thus extending Gross hypercontractivity theory. We provide criteria for these Sobolev type inequalities…

We extend some classical results of Cowling and Meda to the noncommutative setting. Let $(T_t)_{t>0}$ be a symmetric contraction semigroup on a noncommutative space $L_p(\mathcal{M}),$ and let the functions $\phi$ and $\psi$ be regularly…

We present a class of potentials $q \colon \mathbb{R}^{n} \to (0,\infty)$ that implies the weighted Schr\"odinger semigroup $\varphi^{-1}\mathrm{e}^{-tH}\varphi$ to map a weighted Lebesgue function space…

We study heat equations $\frac{\partial u}{\partial t} - \operatorname{div} \left( A \nabla u \right) = 0$ on bounded Lipschitz domains $\Omega$ in $\mathbb{R}^{d}$ for $d \in \mathbb{N}$, where $-\operatorname{div} \left( A \nabla \cdot…

The continuum $\varphi^4_2$ and $\varphi^4_3$ measures are shown to satisfy a log-Sobolev inequality uniformly in the lattice regularisation under the optimal assumption that their susceptibility is bounded. In particular, this applies to…