Related papers: A note on spectral gap and weighted Poincar\'e ine…
The problem of diffusion in a porous medium with a spatially varying porosity is considered. The particular microstructure analyzed comprises a collection of impenetrable spheres, though the methods developed are general. Two different…
We explore in detail the physics potential of a measurement of the ttbar invariant mass distribution. First, we assess the accuracy of the best available predictions for this observable and find that in the low invariant mass region, the…
In a complete metric space that is equipped with a doubling measure and supports a Poincar\'e inequality, we study strict subsets, i.e. sets whose variational capacity with respect to a larger reference set is finite, in the case $p=1$.…
In the first part of these notes, we review some of the recent developments in the study of the spectral properties of Wigner matrices. In the second part, we present a new proof of a Wegner estimate for the eigenvalues of a large class of…
We establish new sharp asymmetric Poincare inequalities in one-dimension with the computation of optimal constants and characterization of extremizers. Using the one-dimensional theory we develop a comprehensive study on fundamental…
In this paper we unify and improve some of the results of Bourgain, Brezis and Mironescu and the weighted Poincar\'e-Sobolev estimate by Fabes, Kenig and Serapioni. More precisely, we get weighted counterparts of the Poincar\'e-Sobolev type…
Experimental methods based on single particle tracking (SPT) are being increasingly employed in the physical and biological sciences, where nanoscale objects are visualized with high temporal and spatial resolution. SPT can probe…
We obtain a sharp estimate for the best constant $C>0$ in the Wirtinger type inequality \[ \int_0^{2\pi}\gamma^pw^2\le C\int_0^{2\pi}\gamma^qw'^2 \] where $\gamma$ is bounded above and below away from zero, $w$ is $2\pi$-periodic and such…
We establish a Shearer-type inequality for the Poincar\'e constant, showing that the Poincar\'e constant corresponding to the convolution of a collection of measures can be nontrivially controlled by the Poincar\'e constants corresponding…
The aim of this paper is to establish various functional inequalities for the convolution of a compactly supported measure and a standard Gaussian distribution on Rd. We especially focus on getting good dependence of the constants on the…
Consider the Poincar\'e-Sobolev inequality on the hyperbolic space: for every $n \geq 3$ and $1 < p \leq \frac{n+2}{n-2},$ there exists a best constant $S_{n,p, \lambda}(\mathbb{B}^{n})>0$ such that $$S_{n, p,…
It is well known from the Perron-Frobenius theory that the spectral gap of a positive square matrix is positive. In this paper, we give a more quantitative characterization of the spectral gap. More specifically, using a complex extension…
We consider a one-dimensional diffusion which solves a stochastic differential equation with Borel-measurable coefficients in an open interval. We allow for the endpoints to be inaccessible or absorbing. Given a Borel-measurable function…
We prove the $L_p$ Poincar\'e inequalities with constant $C\sqrt{p}$ for $1$-cocycles on countable discrete groups under Bakry--Emery's $\Gamma_2$-criterion. These inequalities determine an analogue of subgaussian behavior for 1-cocycles.…
We study geometric characterizations of the Poincar\'{e} inequality in doubling metric measure spaces in terms of properties of separating sets. Given a couple of points and a set separating them, such properties are formulated in terms of…
This is a survey of accumulated spectral analysis observations spanning more than a century, referring to the double layer potential integral equation, also known as Neumann-Poincar\'e operator. The very notion of spectral analysis has…
Traditionally, systems governed by linear Partial Differential Equations (PDEs) are spatially discretized to exploit their algebraic structure and reduce the computational effort for controlling them. Due to beneficial insights of the PDEs,…
The classical Onofri inequality in the two-dimensional sphere assumes a natural form in the plane when transformed via stereographic projection. We establish an optimal version of a generalization of this inequality in the d-dimensional…
We consider probability measures supported on a finite discrete interval $[0,n]$. We introduce a new finitedifference operator $\nabla_n$, defined as a linear combination of left and right finite differences. We show that this operator…
We introduce higher-order Poincar'e constants for compact weighted manifolds and estimate them from above in terms of subsets. These estimates imply upper bounds for eigenvalues of the weighted Laplacian and the first nontrivial eigenvalue…