Related papers: An operator-coefficients free Poincar\'e inequalit…
We develop a Fourier analysis for a generalization of the class of periodic functions, often referred to as $(\theta, T)$-periodic functions, and prove several properties and inequalities related to the Fourier transform, including a type…
In this paper, we characterize the sharp constant and maximizing functions for weighted Poincar\'e inequalities. These results lead to refinements of Hardy's inequality obtained by adding remainder terms involving \(L^p\) norms. We use…
The main aim of this paper is to prove a generalization of the classical Bohr theorem and as an application, we obtain a counterpart of Bohr theorem for the generalized Ces\'aro operator.
Consider the linear stochastic evolution equation dU(t) = AU(t) + dW_H(t), t\ge 0, where A generates a C_0-semigroup on a Banach space E and W_H is a cylindrical Brownian motion in a continuously embedded Hilbert subspace H of E. Under the…
It is known that the Poincar\'e inequality is equivalent to the quadratic transportation-variance inequality (namely $W_2^2(f\mu,\mu) \leqslant C_V \mathrm{Var}_\mu(f)$), see Jourdain \cite{Jourdain} and most recently Ledoux…
We introduce function spaces for the treatment of non-linear parabolic equations with variable $\log$-H\"older continuous exponents, which only incorporate information of the symmetric part of a gradient. As an analogue of Korn's inequality…
We prove that a probability measure on an abstract metric space satisfies a non trivial dimension free concentration inequality for the $\ell_2$ metric if and only if it satisfies the Poincar\'e inequality.
This is a review of some coordinate-free calculi of pseudodifferential operators developed in the last years. As an application, we use a coordinate-free calculus to obtain new results on the behaviour of the spectral projections of a…
In this work we give a sufficient condition under which the global Poincar\'{e} inequality on Carnot groups holds true for a large family of probability measures absolutely continuous with respect to the Lebesgue measure. The density of…
In this short communication, we present a new proof for the Korn inequality in a n-dimensional context. The results are based on standard tools of real and functional analysis. For the final result the standard Poincar\'{e} inequality plays…
The aim of this note is to show that Poincar\'e inequalities imply corresponding weighted versions in a quite general setting. Fractional Poincar\'e inequalities are considered, too. The proof is short and does not involve covering…
Let ${\mathfrak A}$ be a $C^*$-algebra, $T$ be a locally compact Hausdorff space equipped with a probability measure $P$ and let $(A_t)_{t\in T}$ be a continuous field of operators in ${\mathfrak A}$ such that the function $t \mapsto A_t$…
We study generalized Poincar\'e inequalities. We prove that if a function satisfies a suitable inequality of Poincar\'e type, then the Hardy-Littlewood maximal function also obeys a meaningful estimate of similar form. As a by-product, we…
The principal aim of this paper is to employ Bessel-type operators in proving the inequality \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{\sin^2 (x)}+\dfrac{1}{4}\int_0^\pi dx \,…
The present paper is mainly concerned with equations involving exponentials of bounded normal operators. Conditions implying commutativity of those normal operators are given. This is carried out without the known $2\pi i$-congruence-free…
In this paper, we consider a problem in calculus of variations motivated by a quantitative isoperimetric inequality in the plane. More precisely, the aim of this article is the computation of the minimum of the variational problem…
We study the spectrum of the Poincar\'e operator in triaxial ellipsoids subject to a constant rotation. As explained in the paper, this mathematical problem is interesting for many physical applications. It is known that the spectrum of…
This work contributes to nonlocal vector calculus as an indispensable mathematical tool for the study of nonlocal models that arises in a variety of applications. We define the nonlocal half-ball gradient, divergence and curl operators with…
We characterize $B$-valued semi-circular system in terms of $B$-valued free probabilistic analogue of Poincar\'{e} inequality. This is a $B$-valued generalization of Biane's theorem \cite[Theorem 5.1]{b03}. Moreover, we prove that…
Our main result is a weighted fractional Poincar\'e-Sobolev inequality improving the celebrated estimate by Bourgain-Brezis-Mironescu. This also yields an improvement of the classical Meyers-Ziemer theorem in several ways. The proof is…