On the limiting behaviour of Sobolev space functions

Question

Let $k$ be an integer such that $k>n/2$, and let $H^k(\mathbb{R}^n)$ denote the usual Sobolev Hilbert space.

Let $f,g\in H^k(\mathbb{R}^n)$.

Is it true that $\displaystyle \lim_{R\rightarrow \infty} \int_{S_R} f\cdot g\; dS_R=0? $

Here $S_R$ denotes the sphere of radius $R$ centered at $0$, and $dS_R$ the area measure on $S_R$.

Answer 1

If $k>n/2$ then you have (continuously) $H^k(\mathbf{R}^d)\hookrightarrow L^\infty(\mathbf{R}^n)$ by Morrey's Theorem. Since compactly supported functions are dense in $H^k(\mathbf{R}^d)$ you get by uniform convergence (thanks to the previous embedding) that elements of $H^k(\mathbf{R}^d)$ tend to $0$ at infinity.