$\left | \int_{\mathbb R^d} \nabla f \nabla g \right| \leq \|\Delta f \|_{L^2} \|g\|_{L^{2}}$?

Question

Let $f, g \in L^{2}$ with $\nabla f, \nabla g \in L^2$

Question: Can we expect

$\left | \int_{\mathbb R^d} \nabla f \nabla g \right| \leq \|\Delta f \|_{L^2} \|g\|_{L^{2}}$?

(Do I need to assume that $f$ or $g$ is smooth with compact support?)

My try: I think,I should use Cauchy-Schwartz, but before this how I should transfer the $\nabla$ of $g$ to $\Delta$ of $f$?