# Integral inequality with partial derivative

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.

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 $$| \frac{\partial}{\partial x_j} L_\eta (\phi(x)x) | = | L_\eta(\frac{\partial}{\partial x_j} \phi(x)x ) | \le ||L_\eta|| \, || \frac{\partial}{\partial x_j} \phi(x)x ||_2 = ||\eta||_2 || \frac{\partial}{\partial x_j} \phi(x)x ||_2$$