The Leray--Adams inequality

In this paper, we establish the following Leray--Adams type inequality on a bounded domain $Ω$ in R⁴ containing the origin, \[ u∈ C_0^∞(Ω), I_4[u,Ω,R] ≤ 1 ∫_Ωexp(c( {|u|}{E_2^β({|x|}R)})^2) dx ≤ C |Ω| \] for some constants $c >0$ and $C >0$, where β≥ 1, R ≥ x∈ Ω |x|, I₄[u,Ω,R]:= ∫_Ω|Δu|² dx - ∫_Ω|u|²|x|⁴ E₁²(|x|R) dx, and E₁(t) = 1-ln t, E₂(t) = ln (eE₁(t)) for t ∈ (0,1]. This extends the Leray--Trudinger inequality recently established by Psaradakis and Spector {PS2015} and Mallick and Tintarev {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 Bᵣ (with center at origin and radius $r >0$) in Rⁿ.