The Leray--Adams inequality

TL;DR

This paper establishes a new Leray--Adams inequality involving exponential integrability for functions with a Laplacian-based energy constraint on bounded domains in four dimensions, extending previous results to higher dimensions and derivatives.

Contribution

It extends the Leray--Trudinger inequality to the case of the Laplacian operator and higher dimensions, providing new exponential integrability results under specific energy constraints.

Findings

Proves a Leray--Adams type inequality in four dimensions.

Extends the inequality to higher dimensions for radial functions.

Provides bounds involving exponential integrals with specific weight functions.

Abstract

In this paper, we establish the following Leray--Adams type inequality on a bounded domain $\Omega$ in $\mathbb R^{4}$ containing the origin, \[ \sup_{u\in C_0^\infty(\Omega), \tilde I_4[u,\Omega,R] \leq 1} \int_\Omega \exp\left(c\left( \frac{|u|}{E_2^{\beta}\left(\frac{|x|}R\right)}\right)^2\right) dx \leq C |\Omega| \] for some constants $c >0$ and $C >0$, where $\beta\geq 1$, $R \geq \sup_{x\in \Omega} |x|$, $ \tilde I_4[u,\Omega,R]:= \int_\Omega |\Delta u|^2 dx - \int_\Omega \frac{|u|^2}{|x|^{4} E_1^2\left(\frac{|x|}R\right)} dx, $ and $E_1(t) = 1-\ln t$, $E_2(t) = \ln (eE_1(t))$ for $t \in (0,1]$. This extends the Leray--Trudinger inequality recently established by Psaradakis and Spector \cite{PS2015} and Mallick and Tintarev \cite{MT2018} to the case of Laplacian operator. In the higher dimensions or higher order derivatives, we prove the Leray--Adams type inequality for radial function on the ball $B_r$ (with center at origin and radius $r >0$) in $\mathbb R^n$.