Let $f\in C_c^\infty(\mathbb R^3)$. I so far did not see why the following inequality holds:
$$\int_{\mathbb R^3} \lvert \Delta f(x) \rvert \, dx \leq C\sum_{i,j} \lVert \partial_{x_i} \partial_{x_j} f \rVert_1.$$
I only got to the point where $$\int_{\mathbb R^3} \lvert \Delta f(x) \rvert \, dx = \int_{\mathbb R^3} \left (\overline{\Delta f(x)} \Delta f(x) \right)^{1/2} \, dx = \int_{\mathbb R^3} \left (\sum_{i,j} \overline{\partial^2_{x_i}f(x)} \partial ^2_{x_j}f(x) \right)^{1/2} \, dx.$$ How does one proceed from here, i.e. how do the mixed derivatives come in?
$|\Delta f(x)|\leq \sum_{i=1}^3 |\partial^2_{x_i} f(x)|$ by the triangle inequality and clearly, $\sum_{i=1}^3 |\partial^2_{x_i} f(x)|\leq \sum_{i,j=1}^3 |\partial_{x_i}\partial_{x_j} f(x)|,$ since the latter sum simply has some new, added terms. Now, apply linearity of integrals.