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If $v_n \to 0$ in $L^2$, does $v_n \to 0$ in $L^8$? Suppose the domain is a compact surface in $\mathbb{R}^n$. For example it could be a sphere.

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    @maximumtag: The only thing that matters about the interval $[0,1/n^4]$ in that example is its measure. You can generalize to other domains.2012-12-09

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