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I have a questions and maybe you are able to assist with this? Let us consider the space $X:=\mathrm{L}^2[0,\pi]$. On $X$ we consider the family of operators $(P(t,s))_{t\geq s}$ defined by $$ P(t,s)f:=\sum_{n=0}^{\infty}{u_n(t,s)\left\langle f,e_n\right\rangle e_n}, $$ where $(e_n)_{n\in\mathbb{N}}$ is an orthonormal basis for $X$. Moreover, the functions $u_n$ are supposed to be the solutions of the initial value problem given by $$ \begin{cases} u''(t)+n^2\alpha(t)u(t)=0,&\quad 0\leq s\leq t\leq \pi,\\ u(s)=0,&\\ u'(s)=1.& \end{cases} $$ Here $\alpha$ is a continuous differentiable function with $\alpha(t)\geq1$. The question now is, how to determine the following space explicitly: $$ \mathscr{D}_2:=\{f\in X:\ (t,s)\mapsto P(t,s)f\ \text{is twice continuously differentiable}\}. $$ I tried to differentate entrywise, but I do not think that this is allowed without any saying. Is it maybe possible to describe $\mathscr{D}_2$ by means of well-know function spaces within $X$?

Thank you very much for your help in advance.

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  • $\begingroup$ What do you mean by differentiable? Is it with respect to $s$ and $t$, or something else? $\endgroup$
    – Igor Khavkine
    Commented Jun 3, 2021 at 9:28
  • $\begingroup$ @IgorKhavkine Good question. I mean that all partial derivatives exist. Does this help? $\endgroup$
    – Daniella Dannell
    Commented Jun 3, 2021 at 10:06
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    $\begingroup$ Which $\frac{\partial}{\partial ?}$ derivatives? The ambiguity is that in addition to $s$ and $t$ you can also differentiate along your spatial coordinate in $X=L^2[0,\pi]$. $\endgroup$
    – Igor Khavkine
    Commented Jun 3, 2021 at 10:12
  • $\begingroup$ @IgorKhavkine I talk about $\frac{\partial}{\partial{s}}$, $\frac{\partial}{\partial{t}}$ as well as $\frac{\partial^2}{\partial{s}\partial{t}}$, $\frac{\partial^2}{\partial{s}^2}$, etc. $\endgroup$
    – Daniella Dannell
    Commented Jun 3, 2021 at 10:23

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