It is a classical result due to Gagliardo and Nirenberg that it 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 know 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!

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!