= H1 L3 interpolation inequality in three dimensions
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{title2=$\|u\|_3\leq C\|u\|_2^{1/2}\|u\|_{H^1}^{1/2}$}
On a bounded <Lipschitz domain> $U\subseteq\mathbb R^3$, the <Sobolev inequality> gives $\|u\|_6\leq C_U\|u\|_{H^1(U)}$. The <Holder inequality> gives
$$
\|u\|_3\leq\|u\|_2^{1/2}\|u\|_6^{1/2}\leq C_U^{1/2}\|u\|_2^{1/2}\|u\|_{H^1(U)}^{1/2}.
$$
The regular-domain hypothesis supplies the $H^1$-to-$L^6$ <Sobolev embedding theorem>; boundedness of an arbitrary open set alone does not provide that embedding for all $H^1$ functions. On $\mathbb R^3$, the homogeneous <Sobolev inequality> instead gives $\|u\|_3\leq C\|u\|_2^{1/2}\|\nabla u\|_2^{1/2}$ for $u\in H^1(\mathbb R^3)$.
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