Related papers: Sharp Quantitative Forms of the Hardy Inequality o…

Let (M,g) be a smooth compact Riemannian manifold without boundary of dimension n>=6. We prove that {align*} \|u\|_{L^{2^*}(M,g)}^2 \le K^2\int_M\{|\nabla_g u|^2+c(n)R_gu^2\}dv_g +A\|u\|_{L^{2n/(n+2)}(M,g)}^2, {align*} for all u\in H^1(M),…

In this paper, we investigate the validity of a quantitative version of stability for the critical Hardy-H\'enon equation \begin{equation*} H(u):=\div(|x|^{-2a}\nabla u)+|x|^{-pb}|u|^{p-2}u=0,\quad u\in D_a^{1,2}(\R^n), \end{equation*}…

We prove a simple sufficient criteria to obtain some Hardy inequalities on Riemannian manifolds related to quasilinear second-order differential operator $\Delta_{p}u := \Div(\abs{\nabla u}^{p-2}\nabla u)$. Namely, if $\rho$ is a…

In the Riemannian setting, every flat Cartan--Hadamard manifold is isometric to Euclidean space, the canonical model that underlies the theory of Sobolev spaces and guarantees the sharpness/rigidity of the Hardy inequality, the uncertainty…

By using a suitable transform related to Sobolev inequality, we investigate the sharp constants and optimizers in radial space for the following weighted Caffarelli-Kohn-Nirenberg-type inequalities: \begin{equation*}…

In this paper, we study Hardy's inequality in a limiting case: $$ \int_{\Omega} |\nabla u |^N dx \ge C_N(\Omega) \int_{\Omega} \frac{|u(x)|^N}{|x|^N \left(\log \frac{R}{|x|} \right)^N} dx $$ for functions $u \in W^{1,N}_0(\Omega)$, where…

Given $N\geq 3,$ we consider the critical Hardy-Sobolev equation $-\Delta u-\frac{\gamma}{|x|^2}u=\frac{|u|^{2^*(s)-2}u}{|x|^s}$ in $\mathbb{R}^N\setminus \{0\},$ where $0<\gamma<\gamma_{H}:=\left(\frac{N-2}{2}\right)^2,\,s\in (0,2)$ and…

In this paper we prove sharp multipolar Hardy-type inequalities in the Riemannian $L^p-$setting for $p\geq 2$ using the method of super-solutions and fundamental results from comparison theory on manifolds, thus generalizing previous…

We study the following version of Hardy-type inequality on a domain $\Omega$ in a Riemannian manifold $(M,g)$: $$ \int{\Omega}|\nabla u|_g^p\rho^\alpha dV_g \geq \left(\frac{|p-1+\beta|}{p}\right)^p\int{\Omega}\frac{|u|^p|\nabla…

Let $ m, n $ be integers such that $ \frac{n}{2} > m \geq 1 $ and let $ (M, g) $ be a closed $ n-$dimensional Riemannian manifold. We prove there exists some $ B \in \mathbb{R} $ depending only on $ (M, g) $, $ m $, and $ n $ such that for…

This paper investigates sharp stability estimates for the fractional Hardy-Sobolev inequality: $$\mu_{s,t}\left(\mathbb{R}^N\right) \left(\int_{\mathbb{R}^N} \frac{|u|^{2^*_s(t)}}{|x|^t} \,{\rm d}x \right)^{\frac{2}{2^*_s(t)}} \leq…

We study the existence/nonexistence and qualitative properties of the positive solutions to the problem \begin{align*} (-\Delta)^s u -\theta\frac{u}{|x|^{2s}}&=u^p - u^q \quad\text{in }\,\, \mathbb{R}^N,\quad u > 0 \quad\text{in }\,\,…

We consider the prescribed mean curvature equation for entire spacelike hypersurfaces in the Lorentz-Minkowski space, namely \begin{equation*} -\operatorname{div}\left(\displaystyle\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)= \rho \quad…

We prove three sharp bounds for solutions to the porous medium equation posed on Riemannian manifolds, or for weighted versions of such equation. Firstly we prove a smoothing effect for solutions which is valid on any Cartan-Hadamard…

In this paper, we consider the Euclidean logarithmic Sobolev inequality \begin{eqnarray*} \int_{\mathbb{R}^d}|u|^2\log|u|dx\leq\frac{d}{4}\log\bigg(\frac{2}{\pi d e}\|\nabla u\|_{L^2(\mathbb{R}^d)}^2\bigg), \end{eqnarray*} where $u\in…

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…

In this paper, we study the quantitative stability of the nonlocal Soblev inequality \begin{equation*} S_{HL}\left(\int_{\mathbb{R}^N}\big(|x|^{-\mu} \ast |u|^{2_{\mu}^{\ast}}\big)|u|^{2_{\mu}^{\ast}}…

Morrey's classical inequality implies the H\"older continuity of a function whose gradient is sufficiently integrable. Another consequence is the Hardy-type inequality $$ \lambda\biggl\|\frac{u}{d_\Omega^{1-n/p}}\biggr\|_{\infty}^p\le…

In this paper we consider a weighted version of one dimensional discrete Hardy's Inequality on half-line with power weights of the form $n^\alpha$. Namely we consider: \begin{equation} \sum_{n=1}^\infty |u(n)-u(n-1)|^2 n^\alpha \geq…