The Leray--Adams inequality

In this paper, we establish the following Leray–Adams type inequality on a bounded domain $\mathrm{\Omega }$ in ${\mathbb{R}}^4$ containing the origin, $$\underset{u\in C_0^{\mathrm{\infty }}(\mathrm{\Omega }),{\tilde{I}}_4[u,\mathrm{\Omega },R]\le 1}{\sup }{\int }_{\mathrm{\Omega }}\exp \left(c{\left(\frac{\mathrm{\mid }u\mathrm{\mid }}{E_2^{\beta }\left(\frac{\mathrm{\mid }x\mathrm{\mid }}{R}\right)}\right)}^2\right)dx\le C\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }$$ for some constants $c>0$ and $C>0$, where $\beta \ge 1$, $R \geq \sup_{x\in \Omega} |x|$,$ \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 ${\mathbb{R}}^n$.