It is a classical result due to Gagliardo and Nirenberg that there exists a constant C such that it holds
$$
||\nabla \psi|| \_{L ^\infty (\mathbb{R}^2)} ^2
\le ||D ^2 \psi|| \_{L ^\infty (\mathbb{R}^2)} ||\psi|| \_{L ^\infty (\mathbb{R}^2)},
$$
for any $ \psi \in C ^2 (\mathbb{R} ^2) $.


My question would be if anyone knows if it is possible to obtain the same result or a similar one with the laplacian instead of the second derivative, namely if it holds that
$$
||\nabla \psi|| \_{L ^\infty (\mathbb{R}^2)} ^2
\le ||\Delta \psi|| \_{L ^\infty (\mathbb{R}^2)} ||\psi|| \_{L ^\infty (\mathbb{R}^2)}.
$$


Any help or reference will be greatly appreciated!



Gagliardo Nirenberg inequality for the laplacian
By the classical regularity theory for the Poisson equation, you have 
$$
 \Vert \nabla \psi \Vert\_{L^\infty (B\_1)} \le C \bigl(\Vert \Delta \psi \Vert\_{L^\infty (B\_2)} + \Vert \psi \Vert\_{L^\infty (B\_2)} \bigr).
$$
See for example Gilbarg and Trudinger, *Elliptic partial differential equations of second order*, 1983, theorem 3.9, where it is proved by the maximum principle. (An alternative would be to use a Green representation formula of the solution in a ball.)
By scaling this gives 
$$ R\Vert \nabla \psi \Vert\_{L^\infty (B\_R)} \le C \bigl(R^2 \Vert \Delta \psi \Vert\_{L^\infty (B\_{2 R})} + \Vert \psi \Vert\_{L^\infty (B\_{2 R})} \bigr).
$$
By translating the estimate, you obtain thus for every $R > 0$,
$$ R\Vert \nabla \psi \Vert\_{L^\infty (\mathbb{R}^n)} \le C \bigl(R^2 \Vert \Delta \psi \Vert\_{L^\infty (\mathbb{R}^n)} + \Vert \psi \Vert\_{L^\infty (\mathbb{R}^n)} \bigr).
$$
By taking $R = \sqrt{\Vert \psi \Vert\_{L^\infty (\mathbb{R}^n)} / \Vert \Delta \psi \Vert\_{L^\infty (\mathbb{R}^n)}}$, we conclude that 
$$
\Vert \nabla \psi \Vert\_{L^\infty (\mathbb{R}^n)} \le 2C \Vert \Delta \psi \Vert\_{L^\infty (\mathbb{R}^n)}^{1/2} \Vert \psi \Vert\_{L^\infty (\mathbb{R}^n)}^{1/2}.
$$


