Related papers: A spectral result for Hardy inequalities

We study the quasilinear elliptic inequality $$ -\Delta_m u - \frac{\mu}{|x|^m}u^{m-1} \geq (I_\alpha*u^p)u^q \quad\mbox{ in }\mathbb{R}^N\setminus \overline B_1, N\geq 1, $$ where $p>0$, $q, \mu \in \mathbb{R}$, $m>1$ and $I_\alpha$ is the…

We prove various equivalent characterisations of the Hardy space $H^p_{\mathcal{L}}(\mathbb{C}^n)$ for $0<p<1$ associated with the twisted Laplacian $\mathcal{L}$ which generalises the result of [MPR81] for the case $p=1$. Using the atomic…

We study Rellich inequalities associated to higher-order elliptic operators in the Euclidean space. The inequalities are expressed in terms of an associated Finsler metric. In the case of half-spaces we obtain the sharp constant while for a…

We establish sparse Hanson-Wright inequalities for quadratic forms of sparse $\alpha$-sub-exponential random vectors with exponent parameter $\alpha\in(0, 2]$. In the regime $0< \alpha\le 1$ we derive a refined inequality that is optimal in…

When studying the weighted Hardy-Rellich inequality in $L^2$ with the full gradient replaced by the radial derivative the best constant becomes trivially larger or equal than in the first situation. Our contribution is to determine the new…

In this work we establish trace Hardy-Sobolev-Maz'ya inequalities with best Hardy constants, for weakly mean convex domains. We accomplish this by obtaining a new weighted Hardy type estimate which is of independent inerest. We then produce…

We consider a non-relativistic quantum particle interacting with a singular potential supported by two parallel straight lines in the plane. We locate the essential spectrum under the hypothesis that the interaction asymptotically…

This paper establishes a bivariate Hardy-Sobolev inequality. Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be an open domain, $s \in (0,2)$, $\alpha > 1$, $\beta > 1$ with $\alpha + \beta = 2^*(s)$, and $\kappa \in \mathbb{R}$. For any…

Let $0<s<1$ and $p>1$ be such that $ps<N$. Assume that $\Omega$ is a bounded domain containing the origin. Staring from the ground state inequality by R. Frank and R. Seiringer we obtain: 1- The following improved Hardy inequality for $p\ge…

The Hardy-Littlewood inequalities for $m$-linear forms on $\ell_{p}$ spaces are stated for $p>m$. In this paper, among other results, we investigate similar results for $1\leq p\leq m.$ Let $\mathbb{K}$ be $% \mathbb{R}$ or $\mathbb{C}$ and…

In this work we extend the theory of the classical Hardy space $H^1$ to the rational Dunkl setting. Specifically, let $\Delta$ be the Dunkl Laplacian on a Euclidean space $\mathbb{R}^N$. On the half-space $\mathbb{R}_+\times\mathbb{R}^N$,…

Let $\Omega$ be a strongly Lipschitz domain of $\reel^n$. Consider an elliptic second order divergence operator $L$ (including a boundary condition on $\partial\Omega$) and define a Hardy space by imposing the non-tangential maximal…

We provide a short proof of a sharp rearrangement estimate for a generalized version of a potential of Wolff--Havin--Maz'ya type. As a consequence, we prove a reduction principle for that integral operators, that is, a characterization of…

In this paper we present a new method of proof of Hardy type inequalities for two-dimensional quantum Hamiltonians with a magnetic field of finite flux. Our approach gives a quantitative lower bound on the best constant in these…

Elliptic estimates in Hardy classes are proved on domains with minimally smooth boundary. The methodology is different from the original methods of Chang/Krantz/Stein.

This paper is devoted to Hardy inequalities concerning distance functions from submanifolds of arbitrary codimensions in the Riemannian setting. On a Riemannian manifold with non-negative curvature, we establish several sharp weighted Hardy…

In this paper, we first obtain the operator norms of the $n$-dimensional Hardy-Littlewood-P\'{o}lya operator $\mathcal{H}$ from weighted Lebesgue spaces $L^p( \mathbb{R} ^n,| x |^{\beta} ) $ to weighted weak Lebesgue spaces…

We provide lower bounds for the sum of the negative eigenvalues of the operator $|\sigma\cdot p_A|^{2s} - C_s/|x|^{2s} + V$ in three dimensions, where $s\in (0, 1]$, covering the interesting physical cases $s = 1$ and $s = 1/2$. Here…

We give a direct proof of the operator valued Hardy-Littlewood maximal inequality for $2<p<\infty$.

This is the continuation of our previous work [5], where we introduced and studied some nonlinear integral equations on bounded domains that are related to the sharp Hardy-Littlewood-Sobolev inequality. In this paper, we introduce some…