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 $S\_R$. 



On the limiting behaviour of Sobolev space functions
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.


