Related papers: Weighted 1-dimensional Orlicz-Poincar\'e inequalit…

In this paper we study the necessary and sufficient conditions on domain for Musielak-Orlicz-Sobolev embedding of the space $W^{1,\Phi(\cdot,\cdot)}(\Omega)$ where $\Phi(x,t):=t^{p(x)}{(\log(e+t))}^{q(x)}.$

In this paper, we establish a condition on weighted graphs with finite measure that guarantees the validity of a global Poincar\'e inequality. This condition can be viewed as a discrete analogue of the criterion introduced by J. Boman in…

We prove a sharp upper bound on convex domains, in terms of the diameter alone, of the best constant in a class of weighted Poincar\'e inequalities. The key point is the study of an optimal weighted Wirtinger inequality.

It is known that by dualizing the Bochner-Lichnerowicz-Weitzenb\"{o}ck formula, one obtains Poincar\'e-type inequalities on Riemannian manifolds equipped with a density, which satisfy the Bakry-\'Emery Curvature-Dimension condition…

We prove some results about the super Poincar\'e inequality (SPI) and its relation to the spectrum of an operator: we show that it can be alternatively written with Orlicz norms instead of L1 norms, and we use this to give an alternative…

For any convex set $\Omega \subset {\mathbb R} ^N$, we provide a lower bound for the inverse of the Poincar\'e constant in $W ^ {1, 1}(\Omega)$: it refines an inequality in terms of the diameter due to Acosta-Duran, via the addition of an…

For the infinite-dimensional extension of the Dirichlet distribution, the super Poincare inequality does not hold based on the result in [14], so we establish the weighted super Poincare inequalities for this measure with respect to two…

In this paper we give a geometric condition which ensures that $(q,p)$-Poincar\'e-Sobolev inequalities are implied from generalized $(1,1)$-Poincar\'e inequalities related to $L^1$ norms in the context of product spaces. The concept of…

Let $V$ be a locally bounded measurable function such that $e^{-V}$ is bounded and belongs to $L^1(dx)$, and let $\mu_V(dx):=C_V e^{-V(x)} dx$ be a probability measure. We present the criterion for the weighted Poincar\'{e} inequality of…

In this paper, we extend the Marcinkiewicz--Zygmund inequality to the setting of Orlicz and Lorentz spaces. Furthermore, we generalize a Kadec--Pe{\l}czy\'nski-type result -- originally established by the first and third authors for $L^p$…

Poincar\'{e}-Sobolev-type inequalities involving rearrangement-invariant norms on the entire $\mathbb{R}^n$ are provided. Namely, inequalities of the type $\|u-P\|_{Y(\mathbb{R}^n)}\leq C\|\nabla^m u\|_{X(\mathbb{R}^n)}$, where $X$ and $Y$…

In this paper we extend the theory of two weight, $A_p$ bump conditions to the setting of matrix weights. We prove two matrix weight inequalities for fractional maximal operators, fractional and singular integrals, sparse operators and…

We establish Sobolev-Poincar\'e inequalities for piecewise $W^{1,p}$ functions over families of fairly general polytopic (thence also shape-regular simplicial and Cartesian) meshes in any dimension; amongst others, they cover the case of…

We study a Riemannian manifold equipped with a density which satisfies the Bakry--\'Emery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). We first…

We examine the problem of the Fourier transform mapping one weighted Lebesgue space into another, by studying necessary conditions and sufficient conditions which expose an underlying geometry. In the necessary conditions, this geometry is…

In this paper, We establish the weighted Poincar\'{e} inequalities and Log-Sobolev inequalities for Cauchy distributions with optimal weight functions.

We establish Zaremba problem for Laplacian and $p$-Laplacian with degenerate weights when the Dirichlet condition is only imposed in a set of positive weighted capacity. We prove weighted Sobolev-Poincar\'{e} inequality with sharp…

For $p>1$, we introduce the cutoff Sobolev inequality on general metric measure spaces, and prove that there exists a metric measure space endowed with a $p$-energy that satisfies the chain condition, the volume regular condition with…

Payne-P\'olya-Weinberger inequalities are known to be exclusive to bounded Euclidean domains with Dirichlet boundary condition. In this paper, we discuss the corresponding inequalities on Riemannian manifolds of dimension $n \geq3$, and we…

We investigate properties of measures in infinite dimensional spaces in terms of Poincar\'e inequalities. A Poincar\'e inequality states that the $L^2$ variance of an admissible function is controlled by the homogeneous $H^1$ norm. In the…