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Let $I$ be a compact interval of $\mathbb{R}$ and $U$ be a bounded subset of $\mathbb{R}^2$.

If $f \in L^\infty(I, W^{1,2}(U))$, what (non-trivial) condition ($L^p$-estimate on $f$ or decay-like estimate on the $W^{1,2}$-norm of $f$) is sufficient to obtain that $f \in L^\infty(I \times U)$?

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  • $\begingroup$ Can you give some idea of what sort of $L^p$ estimates you have in mind? $\endgroup$
    – Nate Eldredge
    Dec 11, 2018 at 23:36
  • $\begingroup$ @NateEldredge Honestly I don't have a precise idea. $\endgroup$
    – user123457
    Dec 12, 2018 at 1:19
  • $\begingroup$ Well, doesn't $W^{1,2}(U)$ contains functions like $\ln(x^2+y^2)$, which should be in every $L^p$ but unbounded? So I don't see how any simple sort of $L^p$ bound is going to help at all. $\endgroup$
    – Nate Eldredge
    Dec 12, 2018 at 1:34
  • $\begingroup$ @NateEldredge That's true. Maybe some decay-like estimate for some $L^p$ norms or the $H^1$ norm could help? $\endgroup$
    – user123457
    Dec 12, 2018 at 1:43
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    $\begingroup$ What is a decay-like condition on a norm? $\endgroup$
    – Hannes
    Dec 12, 2018 at 10:37

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