# An inequality in Constantin's Navier-Stokes Equations book

The inequality is listed in line 10 on page 29: $\parallel \frac{\partial p}{\partial x_{k}} \parallel_{L^{2}(G_{R^{"}})} \leq c \parallel\nabla \frac{\partial p}{\partial x_{k}} \parallel_{H^{-1}(G_{R})}$. First I would like to know how $\nabla \partial p / \partial x_{k}$ is defined as a functional in $H^{1}_{0}$. Secondly the proof of the inequality. Thanks in advance.

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What is the domain of the $L^2$ norm in the left hand side? –  timur Dec 6 '11 at 5:55