The Leray--Adams inequality

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 $\mathbb R^n$.