Exercise 15, § 5

Let $n\in \mathbf{N}$ and let Δ be the unique Laplacian on $L^2(\mathbf{R}^n)$. Let B denote the unit ball in $\mathbf{R}^n$.

If $g\in \mathscr{L}_u(\mathbf{R}^n)$ is such that the domain of Δ is contained in the domain of $m_g$, then

sup$_{y\in\mathbf{R}}\int^{^n}_B|g(x+y)|^2dx$

is finite. (First verify that $m_g$ is Δ-bounded.)