The Leray--Adams inequality

Abstract

In this paper, we establish the following Leray--Adams type inequality on a bounded domain $\Omega$ in ${\mathbb{R}}^4$ containing the origin, \[ \sup_{u\in C_0^\infty(Ω), \tilde I_4[u,Ω,R] \leq 1} \int_Ω\exp\left(c\left( \frac{|u|}{E_2^β\left(\frac{|x|}R\right)}\right)^2\right) dx \leq C |Ω| \] for some constants $c>0$ and $C>0$, where $\beta \ge 1$, $R\ge {\sup }_{x\in \Omega }|x|$, ${\tilde{I}}_4[u,\Omega ,R]:={\int }_{\Omega }|\Delta u{|}^{2}dx-{\int }_{\Omega }\frac{|u{|}^2}{|x{|}^4{E}_1^2\left(\frac{|x|}{R}\right)}dx,$ and $E_1(t)=1-\ln t$, $E_2(t)=\ln (e{E}_1(t))$ 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 ${\mathbb{R}}^n$.

37 pages, comments are welcome

References in corpus (3)

• Sharp Moser-Trudinger inequalities on Riemannian manifolds with Negative curvature
• On a weighted Trudinger-Moser type inequality on the whole space and related maximizing problem
• Adams Inequality on Pinched Hadamard Manifolds