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 $f$?

  • $\begingroup$ Recall the adjoint of the gradient is the negative of the divergence operator. See math.stackexchange.com/questions/171556/… for details. Then you have $\langle \nabla f,\nabla g\rangle = \langle \Delta f, g\rangle$ and the rest is easy. $\endgroup$ Feb 27, 2017 at 5:00
  • 1
    $\begingroup$ Integrating by parts and use the fact that $L^p$ functions vanish at infinity ($C^{\infty}_0$ being dense in it). $\endgroup$
    – Shuhao Cao
    Feb 27, 2017 at 5:03


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