Related papers: Hardy, Rellich and Uncertainty principle inequalit…

We give the full solution of the following problem: obtain sharp inequalities between the moduli of smoothness $\omega_\alpha(f,t)_q$ and $\omega_\beta(f,t)_p$ for $0<p<q\le \infty$. A similar problem for the generalized $K$-functionals and…

We study sharp second order inequalities of Caffarelli-Kohn-Nirenberg type in the euclidian space $\mathbb{R}^{N}$, where $N$ denotes the dimension. This analysis is equivalent to the study of uncertainty principles for special classes of…

Hardy's type uncertainty principle on connected nilpotent Lie groups for the Fourier transform is proved. An analogue of Hardy's theorem for Gabor transform has been established for connected and simply connected nilpotent Lie groups.…

We prove a Hardy-type inequality for the gradient of the Heisenberg Laplacian on open bounded convex polytopes on the first Heisenberg Group. The integral weight of the Hardy inequality is given by the distance function to the boundary…

In this article we consider linear operators satisfying a generalized commutation relation of a type of the Heisenberg-Lie algebra. It is proven that a generalized inequality of the Hardy's uncertainty principle lemma follows. Its…

We give a new proof of the $L^2$ version of Hardy's uncertainty principle based on calculus and on its dynamical version for the heat equation. The reasonings rely on new log-convexity properties and the derivation of optimal Gaussian decay…

This paper is devoted to show that the flatness of tangents of $1$-codimensional measures in Carnot Groups implies $C^1_\mathbb{G}$-rectifiability. As applications we prove that measures with $(2n+1)$-density in the Heisenberg groups…

This article is concerned with a class of elliptic equations on Carnot groups depending of one real positive parameter and involving a critical nonlinearity. As a special case of our results we prove the existence of at least one nontrivial…

We present a generic functional inequality on Riemannian manifolds, both in additive and multiplicative forms, that produces well known and genuinely new Hardy-type inequalities. For the additive version, we introduce Riccati pairs that…

We investigate the quantitative stability of the nonlocal Sobolev inequality in Heisenberg group \begin{equation*}\label{non-Sobolev} C_{HL}(Q,\mu)…

The classical uncertainty principles deal with functions on abelian groups. In this paper, we discuss the uncertainty principles for finite index subfactors which include the cases for finite groups and finite dimensional Kac algebras. We…

After works by Michael and Simon [10], Hoffman and Spruck [9], and White [14], the celebrated Sobolev inequality could be extended to submanifolds in a huge class of Riemannian manifolds. The universal constant obtained depends only on the…

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}}…

This paper is a second one following our work [CLZ13] in series, considering sharp Hardy- Littlewood-Sobolev inequalities on groups of Heisenberg type. The first important breakthrough was made by Frank and Lieb in [FL12]. In this paper,…

Using a method of factorization and by introducing a generalized discrete Dirichlet's Laplacian matrix $(-\Delta_{\Lambda})$, we establish an extended improved discrete Hardy's inequality and Rellich inequality in one dimension. We prove…

We present a generalization of Hirschman's entropic uncertainty principle for locally compact abelian groups to unimodular locally compact quantum groups. As a corollary, we strengthen a well-known uncertainty principle for compact groups,…

For $N\geq 5$ and $0<\mu<N-4$, we first show a non-degenerate result of the extremal functions for the following Rellich-Sobolev type inequality \begin{align*} \int_{\mathbb{R}^N}|\Delta u|^2 \mathrm{d}x…

Let M be an N-function satisfying the $\Delta_2$- condition, let $\omega, \vp$ be two other functions, $\omega\ge 0$. We study Hardy-type inequalities \[ \int_{\rp} M(\omega (x)|u(x)|) {\rm exp}(-\vp (x))dx \le C\int_{\rp} M(|u'(x)|) {\rm…

We investigate necessary and sufficient conditions on the weights for the Hardy-Rellich inequalities to hold, and propose a new way to use the notion of Bessel pair to establish the optimal Hardy-Rellich type inequalities. Our results…

We prove that if u is a weak solution to a constant coefficient system (with strong ellipticity assumed along the horizontal direction) in a Carnot group (no restriction on the step), then u is actually smooth. We then use this result to…