H1 L3 interpolation inequality in three dimensions ($\Vert u{\Vert }_3\le C\Vert u{\Vert }_2^{1/2}\Vert u{\Vert }_{H^1}^{1/2}$)

On a bounded Lipschitz domain $U\subseteq {\mathbb{R}}^3$, the Sobolev inequality gives $\Vert u{\Vert }_6\le C_U\Vert u{\Vert }_{H^1(U)}$. The Holder inequality givesThe 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 $\Vert u{\Vert }_3\le C\Vert u{\Vert }_2^{1/2}\Vert \nabla u{\Vert }_2^{1/2}$ for $u\in H^1({\mathbb{R}}^3)$.