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

In this paper we discuss the structure of weighted weak Lebesgue spaces and weighted weak Orlicz spaces on $\mathbb{R}^n$. First, we present sufficient and necessary conditions for inclusion relation between weighted weak Lebesgue spaces.…

We consider complete Riemannian manifolds which satisfy a weighted Poincar\`e inequality and have the Ricci curvature bounded below in terms of the weight function. When the weight function has a non-zero limit at infinity, the structure of…

Answering to a recent question raised by Le\'{s}nik, Maligranda, and Tomaszewski, we prove that there is an Orlicz function $\Phi$ with the upper Matuszewska-Orlicz index equal to $1$ such that the Orlicz space $L_\Phi$ does not satisfy…

We study manifolds satisfying a weighed Poincare inequality, which was first introduced by Li-Wang. We generalized one of their results by relaxing the Ricci curvature bound condition only being satisfied outside a compact set and…

In this paper we prove discrete Poincar\'e inequalities that are uniform in the mesh size for the discrete de Rham complex of differential forms developed in [Bonaldi, Di Pietro, Droniou, and Hu, An exterior calculus framework for polytopal…

We present new results on the two-weighted boundedness of singular integral operators and $L^p$ boundedness of the Orlicz maximal function. Namely, we extend a theorem of P\'erez regarding the necessary and sufficient conditions for the…

In this paper, we consider Poincar\'e inequalities for non euclidean metrics on $\mathbb{R}^d$. These inequalities enable us to derive precise dimension free concentration inequalities for product measures. This technique is appropriate for…

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…

We prove that complete Riemannian manifolds with polynomial growth and Ricci curvature bounded from below, admit uniform Poincar\'e inequalities. A global, uniform Poincar\'e inequality for horospheres in the universal cover of a closed,…

In this paper, combining the $p$-capacity for $p\in (1, n)$ with the Orlicz addition of convex domains, we develop the $p$-capacitary Orlicz-Brunn-Minkowski theory. In particular, the Orlicz $L_{\phi}$ mixed $p$-capacity of two convex…

In this short note, we provide a quantitative global Poincar\'e inequality for one forms on a closed Riemannian four manifold, in terms of an upper bound on the diameter, a positive lower bound on the volume, and a two-sided bound on Ricci…

In \cite{MR447956}, Muckenhoupt and Wheeden formulated a weighted weak $(p,p)$ inequality where the weight for the weak $L^p$ space is treated as a multiplier rather than a measure. They proved such inequalities for the Hardy-Littlewood…

We introduce and study the conical curvature-dimension condition, $CCD(K,N)$, for graphs. We show that $CCD(K,N)$ provides necessary and sufficient conditions for the underlying graph to satisfy a sharp global Poincar\'e inequality which in…

We consider weighted norm inequalities for the Riesz potentials $I_\alpha$, also referred to as fractional integral operators. First we prove mixed $A_p$-$A_\infty$ type estimates in the spirit of [13, 15, 17]. Then we prove strong and weak…

We provide necessary and sufficient conditions for the coincidence, up to equivalence of the norms, between strong and weak Orlicz spaces. Roughly speaking, this coincidence holds true only for the so-called exponential spaces. We find also…

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…

We prove a fractional version of Poincar\'e inequalities in the context of $\R^n$ endowed with a fairly general measure. Namely we prove a control of an $L^2$ norm by a non local quantity, which plays the role of the gradient in the…

The main result of this paper supports a conjecture by C. P\'erez and E. Rela about a very recent result of theirs on self-improving theory. Also, we extend the conclusions of their theorem to the range $p<1$. As an application of our…

We extend the results of [5], where we proved an equivalence between weighted Poincar\'e inequalities and the existence of weak solutions to a family of Neumann problems related to a degenerate $p$-Laplacian. Here we prove a similar…

This paper extends the self-improvement result of Keith and Zhong in [16] to the two-measure case. Our main result shows that a two-measure $(p,p)$-Poincar\'e inequality for $1<p<\infty$ improves to a $(p,p-\varepsilon)$-Poincar\'e…