Consider $f \in L^{2}(R^n)$ with $\Delta f \in L^{2}(R^n) $. Show that ${\partial}^{|\alpha| } , (|\alpha| \leq 2 )f \in L^{2}(R^n)$. (the derivatives is in the distribution sense).
My book gives the following hint: $0 \leq (1 - |\theta_j|)^2 , j=1,...,n$ and $0 \leq [1- (|\theta_j| - |\theta_k|)]^2 ,j,k = 1,..,n$ .
But i dont have idea how to start, someone can give me a hint to this exercise ?
Thanks in advance
By Parseval, we have $\hat f\in L^2$ and $|\cdot|^2 \hat f\in L^2$, because the Laplacian becomes the multiplication by $|\xi|^2$ in the Fourier space. Since $$ 2|\xi_j\xi_k| \leq |\xi_j|^2+|\xi_k|^2 \leq |\xi|^2, $$ and the mixed derivative $\partial_j\partial_k$ becomes the multiplication by $\xi_j\xi_k$, we conclude that $\partial_j\partial_k f\in L^2$ as well. I leave the first order derivatives to you.