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Let $ \Omega \in R^N$ be a limited open set, $N \ge 3$, and $p \in (1,\alpha]$ where $\alpha=\frac{2N}{N-2}$, $\phi:\Omega \rightarrow R$. Let $\phi \in H^{2,p'}(\Omega)$ where $\frac{1}{p} + \frac{1}{p'} =1$, and suppose $\phi^+ \in H^1_0(\Omega)$, where $\phi^+$ is the positive part of $\phi$.

Is it true that $\phi \in L^p$ and that $\phi \in H^{1,2}$?

Rocc_00
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    The community appreciates your efforts to use formatting in mathjax. – insipidintegrator Sep 23 '22 at 13:15
  • Note that if $\phi \in H^{1,2}$, then certainly $\phi \in L^p$ for all $p \le 2^\ast$. Next consider the case where $p = 2^\ast$ and $p' = 2N/(N+2)$. Can you prove that $\phi \in H^{1,2}$? – Hans Engler Sep 23 '22 at 13:38
  • Welcome. MathJax use is indeed appreciated, but so are good titles. Having a good title benefits you, as it means those knowledgeable about / interested in your question can see at a glance that they want to read your post, and make it gain attention / answers / useful comments. But as it stands, "$L^p$" is too broad a topic – FShrike Sep 23 '22 at 16:03
  • Hans Engler is it possible to prove it? – Rocc_00 Sep 23 '22 at 22:22
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    What is your definition of $H^{s,p}$ and what is $\phi^+$ ? – LL 3.14 Sep 23 '22 at 22:42
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    $H^{s,p}$ is the space of functions in $L^p$ with weak derivative in $L^p$ until order s. $\phi^ +$ is the positive part of the function $\phi$ – Rocc_00 Sep 24 '22 at 11:00

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