Fix $\eta \in \Bbb R^n$, then for any $\phi \in C_c^\infty(\Bbb R^n)$, i.e. compactly supported smooth function, $$ \int_{\Bbb R^n} \frac{\partial}{\partial x_j} (\phi(x) (x \eta)) \, dx \le \int_{\Bbb R^n} |\eta| \Big| \frac{\partial}{\partial x_j} (\phi(x)x) \Big| \, dx $$ Here $x \eta $ denotes dot product.

Any hints/ advice?

EDIT: I think this is correct.

Consider the bounded operator, $L_\eta:\Bbb R^n \rightarrow \Bbb R^n$, $x \mapsto \langle x, \eta \rangle $. Its norm is given by $||\eta||_2$ by Hilbert space duality.By continuity, we have $$| \frac{\partial}{\partial x_j} L_\eta (\phi(x)x) | = | L_\eta(\frac{\partial}{\partial x_j} \phi(x)x ) | \le ||L_\eta|| \, || \frac{\partial}{\partial x_j} \phi(x)x ||_2 = ||\eta||_2 || \frac{\partial}{\partial x_j} \phi(x)x ||_2 $$


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