Related papers: Ultracontractivity and functional inequalities on …

We construct a family of infinite-dimensional reduced Heisenberg groups which can be viewed as infinite-dimensional homogeneous spaces. Such a space is an analogue of finite-dimensional reduced Heisenberg groups in infinite dimensions. We…

We introduce a class of non-commutative Heisenberg like infinite dimensional Lie groups based on an abstract Wiener space. The Ricci curvature tensor for these groups is computed and shown to be bounded. Brownian motion and the…

A family of logarithmic Sobolev inequalities on finite dimensional quantum state spaces is introduced. The framework of non-commutative $\bL_p$-spaces is reviewed and the relationship between quantum logarithmic Sobolev inequalities and the…

The hypercontractivity is proved for the Markov semigroup associated to a class of finite/infinite dimensional stochastic Hamiltonian systems. Consequently, the Markov semigroup is exponentially convergent to the invariant probability…

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…

We show that the $L^p$ boundedness, $p>2$, of the Riesz transform on a complete non-compact Riemannian manifold with upper and lower Gaussian heat kernel estimates is equivalent to a certain form of Sobolev inequality. We also characterize…

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…

Based on earlier work by Carlen-Maas and the second- and third-named author, we introduce the notion of intertwining curvature lower bounds for graphs and quantum Markov semigroups. This curvature notion is stronger than both Bakry-\'Emery…

Let $G=(V, E)$ be a locally finite connected graph satisfying curvature-dimension conditions ($CDE(n, 0)$ or its strengthened version $CDE'(n, 0))$) and polynomial volume growth conditions of degree $m$. We systematically establish sharp…

We derive sharp, local and dimension dependent hypercontractive bounds on the Markov kernel of a large class of diffusion semigroups. Unlike the dimension free ones, they capture refined properties of Markov kernels, such as trace…

The classical hypercontractive inequality for the noise operator on the discrete cube plays a crucial role in many of the fundamental results in the Analysis of Boolean functions, such as the KKL (Kahn-Kalai-Linial) theorem, Friedgut's…

We prove the sharp Gaussian isoperimetric inequality for conjugate heat-kernel measures along a Ricci flow via a monotonicity formula. As consequences, we obtain the exact Gaussian enlargement theorem and a Gaussian-quantile two-set…

For a function $f$ on the hypercube $\{0,1\}^n$ with Fourier expansion $f=\sum_{S\subseteq[n]}\hat f(S)\chi_S$, the hypercontractive inequality allows bounding norms of $T_\rho f=\sum_S\rho^{|S|} \hat f(S)\chi_S$ in terms of norms of $f$.…

Let $\alpha\in(0,2)$ and $d\in{\mathbb N}$. Consider the following SDE in ${\mathbb R}^d$:$${\rm d}X_t=b(t,X_t){\rm d} t+a(t,X_{t-}){\rm d} L^{(\alpha)}_t,\ \ X_0=x,$$where $L^{(\alpha)}$ is a $d$-dimensional rotationally invariant…

We continue to establish uniform upper bounds and asymptotic expansions for the kernels of the index transforms which were recently developed for the Kontorovich-Lebedev operator. It involves the Mehler-Fock, Lebedev, index Whittaker and…

We present a direct proof of the cutoff Sobolev inequality on the Sierpi\'nski gasket, which has long been regarded as highly non-trivial in the context of heat kernel estimates.

We derive upper bounds for the trace of the heat kernel $Z(t)$ of the Dirichlet Laplace operator in an open set $\Omega \subset \R^d$, $d \geq 2$. In domains of finite volume the result improves an inequality of Kac. Using the same methods…

In sub-Riemannian geometry there exist, in general, no known explicit representations of the heat kernels, and these functions fail to have any symmetry whatsoever. In particular, they are not a function of the control distance, nor they…

We analyze the heat kernel associated to the Laplacian on a compact metric graph, with standard Kirchoff-Neumann vertex conditions. An explicit formula for the heat kernel as a sum over loops, developed by Roth and Kostrykin, Potthoff, and…

We introduce heat semigroup-based Besov classes in the general framework of Dirichlet spaces. General properties of those classes are studied and quantitative regularization estimates for the heat semigroup in this scale of spaces are…