arXiv research

For negatively curved manifolds, a condition is found for intrinsic ultracontractivity of heat semigroups.

In this paper, we prove the equivalent of ultracontractive bound of heat semigroup or the uniform upper bound of the heat kernel with the Nash inequality, Log-Sobolev inequalities on graphs. We also show that under the assumption of volume growth and nonnegative curvature $CDE'(n,0)$ the Sobolev inequality, Nash inequa…

We derive new estimates for the first Betti number of compact Riemannian manifolds. Our approach relies on the Birman-Schwinger principle and Schatten norm estimates for semigroup differences. In contrast to previous works we do not require any a priori ultracontractivity estimates and we provide bounds which explicitl…

We prove a global Li-Yau inequality for a general Markov semigroup under a curvature-dimension condition. This inequality is stronger than all classical Li-Yau type inequalities known to us. On a Riemannian manifold, it is equivalent to a new parabolic Harnack inequality, both in negative and positive curvature, giving…

We consider operators of the form ${\mathcal L}=-L-V$, where $L$ is an elliptic operator and $V$ is a singular potential, defined on a smooth bounded domain $Ω\subset \R^n$ with Dirichlet boundary conditions. We allow the boundary of $Ω$ to be made of various pieces of different codimension. We assume that ${\mathcal L…

Let $P_t$ be the diffusion semigroup generated by $L:=Δ+\nabla V$ on a complete connected Riemannian manifold with $\operatorname {Ric}\ge-(σ^2ρ_o^2+c)$ for some constants $σ, c>0$ and $ρ_o$ the Riemannian distance to a fixed point. It is shown that $P_t$ is hypercontractive, or the log-Sobolev inequality holds for the…

Proves upper bounds for heat kernels evolving on manifolds.

The article examines entropy-information inequalities for continuous-time Markov chains under curvature-dimension conditions.

We establish new Calderón reproducing formulas for self-adjoint operators $D$ that generate strongly continuous groups with finite propagation speed. These formulas allow the analysing function to interact with $D$ through holomorphic functional calculus whilst the synthesising function interacts with $D$ through funct…

Study on Wasserstein gradient flow for MMD between Coulomb measures.