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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 $S_R$.

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Yes. If $\int_{\partial B_r} |f|^2 \, d\sigma > c_0$ on a sequence of slices $r_k$ tending to infinity, then using the $H^k$ norm (or $C^\alpha$ norm, since you imposed $k > n/2$), you can show that the integral of $|f|^2$ must be greater than $c_1$ on a sequence of fattened slices. This violates that $f$ is globally $L^2$. More generally, similar results hold as long as $f$ has some positive regularity in the radial variables, to control the behavior between slices.

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    $\begingroup$ (Also, this is not quite a research-level question.) $\endgroup$ – sharpend Jul 5 at 17:27

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