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I have a question about the time regularity of the traces in one dimension.

Suppose I have a function space $$X = C^1([0,T];L^2(0,1))\cap C([0,T],H^1(0,1))$$ and I define an operator $E$ on $X$ by $(Ef)(t) = f(0,t).$ I know that $E$ maps $X$ to $C([0,T]).$

What can we say about $\partial_t Ef$? Where would that function live? What would be the minimal conditions on $X$ so I would have $Ef \in C^1([0,T])$.

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  • $\begingroup$ In "$f(0,t)$" where do the 0 and $t$ live? In $[0,T]$ and $(0,1)$? Which is which? $\endgroup$
    – Nik Weaver
    Commented Feb 10, 2022 at 12:32
  • $\begingroup$ I am using $t$ for the time variable and $x$ for the space variable. Then since $H^1(0,1)$ embeds into $C([0,1])$ I defining $f(0,t)$ (the trace), so $0 \in [0,1]$ and $t \in [0,T].$ Thanks for the help :) $\endgroup$
    – TOT
    Commented Feb 10, 2022 at 12:42

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