Related papers: Poincar\'e type and spectral gap inequalities with…
We introduce a spectrum of monotone coarse invariants for metric measure spaces called Poincar\'{e} profiles. The two extremes of this spectrum determine the growth of the space, and the separation profile as defined by…
We prove inequalities relating the degrees of holomorphic distributions and of holomorphic foliations forming a flag on $\mathbb{P}^n$. Such inequalities are inspired by the so called Poincar\'e problem for foliations.
Given a complete Riemannian manifold satisfying a weighted Poincar\'{e} inequality and having a bounded below Ricci curvature, various vanishing theorems for harmonic functions and harmonic 1-forms have been published. We generalized these…
We give bounds on the Poincare (inverse spectral gap) constant of a non-negative, integer-valued random variable W, under negative dependence assumptions such as ultra log-concavity and total negative dependence. We show that the bounds…
We consider generalised Mehler semigroups and, assuming the existence of an associated invariant measure $\sigma$, we prove functional integral inequalities with respect to $\sigma$, such as logarithmic Sobolev and Poincar\'{e} type.…
Spaces of differential forms over configuration spaces with Poisson measures are constructed. The corresponding Laplacians (of Bochner and de Rham type) on 1-forms and associated semigroups are considered. Their probabilistic interpretation…
We prove Hardy-type inequalities for a fractional Dunkl--Hermite operator which incidentally give Hardy inequalities for the fractional harmonic oscillator as well. The idea is to use $h$-harmonic expansions to reduce the problem in the…
We prove a Poincar\'e-Sobolev type inequality on compact Riemannian manifolds where the deviation of a function from a biased average, defined using a density, is controlled by the unweighted Lebesgue norm of its gradient. Unlike classical…
In this paper, we will use the normalized intetral Ricci curvature to investigate Liouville type property of $ p $ harmonic function on Riemannian manifold. secondly, we will use the BiRic curvature to obtian Liuville theorem for $ p $…
We present two extensions of the one dimensional free Poincar\'e inequality similar in spirit to two classical refinements.
We investigate the validity, as well as the failure, of Sobolev-type inequalities on Cartan-Hadamard manifolds under suitable bounds on the sectional and the Ricci curvatures. We prove that if the sectional curvatures are bounded from above…
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…
We prove a number of unconditional statistical results of the Hecke coefficients for unitary cuspidal representations of $\operatorname{GL}(2)$ over number fields. Using partial bounds on the size of the Hecke coefficients, instances of…
Firstly, we derive in dimension one a new covariance inequality of $L_{1}-L_{\infty}$ type that characterizes the isoperimetric constant as the best constant achieving the inequality. Secondly, we generalize our result to $L_{p}-L_{q}$…
We establish a general scale-dependent Poincar\'{e}-Hardy type identity involving a vector field on the hyperbolic space. By choosing suitable parameter, potential and vector field in this identity, we can recover, as well as derive new…
We prove a uniform boundary Harnack inequality for nonnegative harmonic functions of the fractional Laplacian on arbitrary open set $D$. This yields a unique representation of such functions as integrals against measures on $D^c\cup…
We study some basic properties of the function $f_0:M\rightarrow\IR$ on Hadamard manifolds defined by $$ f_0(x):=\langle u_0,\exp_{x_0}^{-1}x\rangle\quad\mbox{for any $x\in M$}. $$ A characterization for the function to be linear affine is…
The infinitesimal forms of the $L_{p}$-Brunn-Minkowski inequalities for variational functionals, such as the $q$-capacity, the torsional rigidity, and the first eigenvalue of the Laplace operator, are investigated for $p \geq 0$. These…
In this paper, the author obtains new estimates on generalization of Hadamard, Ostrowski and Simpson type inequalities for Lipschitzian functions via Hadamard fractional integrals. Some applications to special means of positive reals…
In this paper some Hadamard-type inequalities for convex functions of 3-variables on a rectanguler box are given. We also define a mapping related to convex functions on a rectanguler box.