Related papers: An Improved Leray-Trudinger Inequality

Let $\Omega \subset \mathbb{R}^n$ be a smooth bounded domain having zero in its interior $0 \in \Omega.$ We fix $0 < \alpha \le 2$ and $0 \le s <\alpha.$ We investigate a sufficient condition for the existence of a positive solution for the…

In the paper, the authors find the best numbers $\alpha$ and $\beta$ such that $$ \overline{C}\bigl(\alpha a+(1-\alpha)b,\alpha b+(1-\alpha)a\bigr)<T(a,b) <\overline{C}\bigl(\beta a+(1-\beta)b,\beta b+(1-\beta)a\bigr) $$ for all $a,b>0$…

We consider an elliptic differential inequality: $\vert \Delta u(x) \vert \le C_0(\YYYY^{-\gamma}\vert u(x)\vert + \YYYY^{-\theta}\vert \nabla u(x)\vert)$ in an exterior domain $\R^n \setminus \ooo{U}$, where $U$ is a simply connected…

We present a Kato-type inequality for bounded domain Omega \subset R^n, n>1.

In this work, we obtain an existence of nontrivial solutions to a minimization problem involving a fractional Hardy-Sobolev type inequality in the case of inner singularity. Precisely, for $\lambda>0$ we analyze the attainability of the…

We prove that the isoperimetric profile of a convex domain $\Omega$ with compact closure in a Riemannian manifold $(M^{n+1},g)$ satisfies a second order differential inequality which only depends on the dimension of the manifold and on a…

Let $\tau_k(\Omega)$ be the $k$-th eigenvalue of the Laplace operator in a bounded domain $\Omega$ of the form $\Omega_{\text{out}} \setminus \overline{B_{\alpha}}$ under the Neumann boundary condition on $\partial \Omega_{\text{out}}$ and…

Let $n \geq 2$ and $\Omega \subset \mathbb{R}^n$ be a bounded domain. Then by Trudinger-Moser embedding, $W_0^{1,n}(\Omega)$ is embedded in an Orlicz space consisting of exponential functions. Consider the corresponding semi linear…

We prove an improvement for the sharp Adams inequality in $W^{m,\frac nm}_0(\Omega)$ where $\Omega$ is a bounded domain in $\mathbb{R}^n$ inspired by Lions Concentration--Compactness principle for the sharp Moser--Trudinger inequality. Our…

This article establishes a universal robust limit theorem under a sublinear expectation framework. Under moment and consistency conditions, we show that, for $\alpha \in(1,2)$, the i.i.d. sequence \[ \left \{ \left(…

Let $d \ge 1$, $p \ge d$, and let $\Omega$ be a smooth bounded open subset of $\mathbb{R}^d$. We prove some exponential integrability in the spirit of Moser-Trudinger's inequalities for measurable functions $u$ defined in $\Omega$ such that…

We prove the sharp inequality \[ J(\Omega) := \frac{\lambda_1(\Omega)}{h_1(\Omega)^2} < \frac{\pi^2}{4},\] where $\Omega$ is any planar, convex set, $\lambda_1(\Omega)$ is the first eigenvalue of the Laplacian under Dirichlet boundary…

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…

We consider the problem of finding the optimal exponent in the Moser-Trudinger inequality \[ \sup \left\{\int_\Omega \exp{\left(\alpha\,|u|^{\frac{N}{N-s}}\right)}\,\bigg|\,u \in…

We fill the gap left open in \cite{MT}, regarding the minimum exponent on the logarithmic correction weight so that the Leray-Trudinger inequality (see \cite{PsSp}) holds. Instead of the representation formula used in \cite{PsSp} and…

Let $S(t) = \frac{1}{\pi}\Im \log\zeta\left(\frac{1}{2}+it\right)$. We prove an unconditional lower bound on the measure of the sets $\{t\in [T,2T] \colon S(t) \geq V\}$ for $\sqrt{\log\log T} \leq V \ll \left(\frac{\log T}{\log \log…

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d\geq 2$. We prove that if $e(\widehat{R}_{red})>1$, then the classical Lech's inequality can be improved uniformly for all $\mathfrak{m}$-primary ideals, that is, there exists…

A classical result of Aubin states that the constant in Moser-Trudinger-Onofri inequality on $\mathbb{S}^{2}$ can be imporved for furnctions with zero first order moments of the area element. We generalize it to higher order moments case.…

The Berezin--Li--Yau and the Kr\"oger inequalities show that Riesz means of order $\geq 1$ of the eigenvalues of the Laplacian on a domain $\Omega$ of finite measure are bounded in terms of their semiclassical limit expressions. We show…

The principal aim of this note is to illustrate how factorizations of singular, even-order partial differential operators yield an elementary approach to classical inequalities of Hardy-Rellich-type. More precisly, introducing the…