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The following is a theorem in the preliminary material section of Constantin and Foias's "Navier-Stokes Equations":

Let $\Omega\subset\Bbb{R}^n$ be an open bounded set with locally Lipschitz boundary.

  • If a distribution $p\in D'(\Omega)$ has all its first derivatives $D_ip$ in $L^2(\Omega)$ then $p\in L^2(\Omega)$ and $$ \|p\|_{L^2(\Omega)/\Bbb{R}}\leq C(\Omega)\|\nabla p\|_{L^2(\Omega)^n} $$

  • If a distribution $p$ has all its first derivatives in $H^{-1}(\Omega)$ then $p\in L^2(\Omega)$ and $$ \|p\|_{L^2(\Omega)/\Bbb{R}}\leq C(\Omega)\|\nabla p\|_{H^{-1}(\Omega)^n} $$

In both cases, if no restriction is imposed on $\partial\Omega$ it follows that $p\in L_{\hbox{loc}}^2(\Omega)$. By $\|p\|_{L^2(\Omega)/\Bbb{R}}$ we mean $$ \inf_{c\in\Bbb{R}}\|p-c\|_{L^2(\Omega)}= \|p-\frac{\int_\Omega p \ dx}{|\Omega|}\cdot 1\|_{L^2(\Omega)}. $$

The space $L^2(\Omega)/\Bbb{R}$ defined in the theorem looks quite strange to me (since the notation usually suggests a quotient space). Could anyone come up with a cited reference about this space?

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It is a quotient space!

$\mathbb{R}$ here should be thought of as the one-dimensional subspace of $L^2(\Omega)$ which consists of the (a.e.) constant functions. The norm given is the canonical norm on the quotient of a Banach space mod a closed subspace. This construction is discussed, for instance, in section III.4 of Conway's A Course in Functional Analysis.

Strictly speaking, $\|p\|_{L^2(\Omega)/ \mathbb{R}}$ is not really the norm of $p$, but the norm of the coset $p + \mathbb{R}$ in the quotient space. It's an easy exercise to check it is well defined.

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