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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 $f$?

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  • Recall the adjoint of the gradient is the negative of the divergence operator. See http://math.stackexchange.com/questions/171556/gradient-operator-the-adjoint-of-minus-divergence-operator for details.

    Then you have $\langle \nabla f,\nabla g\rangle = \langle \Delta f, g\rangle$ and the rest is easy.

     – Adam Hughes Feb 27 '17 at 05:00
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    Integrating by parts and use the fact that $L^p$ functions vanish at infinity ($C^{\infty}_0$ being dense in it). – Shuhao Cao Feb 27 '17 at 05:03

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