# On the limiting behaviour of Sobolev space functions

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 $$S_R$$.

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.