相关论文: The sharp higher order Lorentz--Poincar\'e and Lor…

Let $W^1L^{p,q}(\mathbb H^n)$, $1\leq q,p < \infty$ denote the Lorentz-Sobolev spaces of order one in the hyperbolic spaces $\mathbb H^n$. Our aim in this paper is three-fold. First of all, we establish a sharp Poincar\'e inequality in…

In this note, we establish a $L^p-$version of the Poincar\'e--Sobolev inequalities in the hyperbolic spaces $\mathbb H^n$. The interest of this result is that it relates both the Poincar\'e (or Hardy) inequality and the Sobolev inequality…

We establish several Poincar\'e--Sobolev type inequalities for the Lapalce--Beltrami operator $\Delta_g$ in the hyperbolic space $\mathbb H^n$ with $n\geq 5$. These inequalities could be seen as the improved second order Poincar\'e…

In this paper, we obtain the sharp $k$-th order Sobolev inequalities in the hyperbolic space ${\H}^n$ for all $k=1,2,3,\cdots$. This gives an answer to an open question raised by Aubin in [5, p.$\;$176-177] for $W^{k,2}({\H}^n)$ with $k>1$.…

By using, among other things, the Fourier analysis techniques on hyperbolic and symmetric spaces, we establish the Hardy-Sobolev-Maz'ya inequalities for higher order derivatives on half spaces. The proof relies on a Hardy-Littlewood-Sobolev…

We prove an improved version of Poincar\'e-Hardy inequality in suitable subspaces of the Sobolev space on the hyperbolic space via Bessel pairs. As a consequence, we obtain a new Hardy type inequality with an improved constant (than the…

In this paper we prove higher order Poincar\'e inequalities involving radial derivatives namely, \begin{equation*} \int_{\mathbb{H}^{N}} |\nabla_{r,\mathbb{H}^{N}}^{k} u|^2 \, {\rm d}v_{\mathbb{H}^{N}} \geq…

In this paper, we prove Poincar\'e and Sobolev inequalities for differential forms in $L^1(\mathbb R^n)$. The singular integral estimates that it is possible to use for $L^p$, $p>1$, are replaced here with inequalities which go back to…

Let $\Omega$ be a bounded John domain in $\mathbb R^n$ with $n\ge 2$, and let $\mathcal{H}_{\infty }^{\delta}$ denote the Hausdorff content of dimension $\delta\in (0,n]$. In this article, the authors prove the Poincar\'e and the…

We investigate several functional and geometric inequalities on the hyperbolic space $\mathbb{H}^N$, with a primary emphasis on logarithmic Sobolev inequalities, Poincar\'e inequalities, and Beckner-type inequalities, all studied within the…

In the Euclidean space $\mathbb{R}^d$, the sharp classical Sobolev inequality is equivalent by conformal invariance to a Sobolev inequality on the hyperbolic space $\mathbb{H}^d$. This inequality is sharp in dimension $d\geq 4$, but it is…

We prove second and fourth order improved Poincar\'e type inequalities on the hyperbolic space involving Hardy-type remainder terms. Since theirs l.h.s. only involve the radial part of the gradient or of the laplacian, they can be seen as…

We study Hardy-type inequalities associated to the quadratic form of the shifted Laplacian $-\Delta_{\mathbb H^N}-(N-1)^2/4$ on the hyperbolic space ${\mathbb H}^N$, $(N-1)^2/4$ being, as it is well-known, the bottom of the $L^2$-spectrum…

We derive sharp Sobolev inequalities for Sobolev spaces on metric spaces. In particular, we obtain new sharp Sobolev embeddings and Faber-Krahn estimates for H\"{o}rmander vector fields.

The paper deals about Hardy-type inequalities associated with the following higher order Poincar\'e inequality: $$ \left( \frac{N-1}{2} \right)^{2(k -l)} := \inf_{ u \in C_{c}^{\infty} \setminus \{0\}} \frac{\int_{\mathbb{H}^{N}}…

We present a unified strategy to derive Hardy-Poincar\'e inequalities on bounded and unbounded domains. The approach allows proving a general Hardy-Poincar\'e inequality from which the classical Poincar\'e and Hardy inequalities immediately…

In this paper we will study the equivalence between super-Poincar\'e inequality and some log-Sobolev type inequalities, including weak log-Sobolev inequality and super log-Sobolev inequality. The explicit relations between associated rate…

The notions of higher-order weighted multilinear Poincar\'e and Sobolev inequalities in Carnot groups are introduced. As an application, weighted Leibnitz-type rules in Campanato-Morrey spaces are established.

We prove several interesting equalities for the integrals of higher order derivatives on the homogeneous groups. As consequences, we obtain the sharp Hardy--Rellich type inequalities for higher order derivatives including both the…

We investigate the possibility of improving the $p$-Poincar\'e inequality $\|\nabla_{\mathbb{H}^N} u\|_p \ge \Lambda_p \|u\|_p$ on the hyperbolic space, where $p>2$ and $\Lambda_p:=[(N-1)/p]^{p}$ is the best constant for which such…