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Consider the warped product $X = M \times \mathbb{R}$, with the metric $g = dr^2 + \varphi(r) g_M$, where $M$ is a compact manifold. Consider the Sobolev space $H^1(X)$ and let $H^1_{rad}(X)$ denote those $H^1$-functions that are also radial. Now, away from the origin, could we say that an element of $H^1_rad(X)$ is actually continuous? This seems to be akin to a Sobolev embedding type statement $H^1_{rad} (X \setminus K) \subset C(X \setminus K)$, where $K$ is a compact region containing the origin. My questions are: does the above inclusion hold in general, or is some kind of restriction on $\varphi(r)$ needed? Also, is the above inclusion continuous? Thanks.

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  • $\begingroup$ I'm voting to close this question because the user account of the OP does not exist any more. $\endgroup$
    – Stefan Kohl ♦
    May 22, 2015 at 13:08
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    $\begingroup$ @StefanKohl If the question is good, does it matter if any "real" entity asked it? (This particular question is outside my subject knowledge, so I cannot tell.) $\endgroup$
    – Boris Bukh
    May 22, 2015 at 13:15
  • $\begingroup$ @StefanKohl I'm with Boris here. With the possible exception of known problem users, why not judge based on the question rather than the user? I know of other examples of legitimate users who have deleted their accounts for one reason or another; it doesn't make their existing question bad. $\endgroup$
    – Ben Webster ♦
    May 23, 2015 at 21:38
  • $\begingroup$ @BenWebster: Why would one ask a question and immediately delete one's account? -- On the other hand, if the question is reasonable, it is reasonable and I retract my close vote. $\endgroup$
    – Stefan Kohl ♦
    May 23, 2015 at 22:31
  • $\begingroup$ @StefanKohl ¯\_(ツ)_/¯ People do strange things from time to time. But if the question is reasonable (I'm also far enough away from the area that I'm agnostic, but of course, moderators have to be especially careful with close votes), then people can still benefit from it, and from reading its answers. $\endgroup$
    – Ben Webster ♦
    May 24, 2015 at 0:56

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