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I am wondering what sort of results are available for the following sort of problem, or where to look in the literature for work dealing with such problems, especially in the degenerate elliptic context. It seems like there should be a general functional-analytic framework, but I'm not sure where to look.

Suppose that one has an order-$N$ (nonlinear) differential operator $P:C^\infty\to C^\infty$ which for any $k\geq N$ extends to a map $C^{k,\alpha}\to C^{k-N,\alpha}$ which is smooth as a map between Banach spaces. Suppose that $P$ has an inverse $Q:C^\infty\to C^\infty$. Suppose that there is $m\in\mathbb{N}$ such that, for any $k\geq 0$, $Q$ has a unique continuous extension to a map $C^{k,\alpha}\to C^{k+m,\alpha}$. Is $Q$ necessarily smooth as a map between these Banach spaces?

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    $\begingroup$ For elliptic operators, smoothness of the solution operator is established in Section II.3.3 of "Nash-Moser Inverse Fuction Theorem" by R. Hamilton (1982). $\endgroup$
    – Tobias Diez
    Commented Apr 18, 2020 at 11:39
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    $\begingroup$ Yes, he only discusses the linear case but for operators depending on parameters. This allows you then to use the Nash-Moser inverse function theorem to conclude that a non-linear solution operator exists (locally) and is smooth. You might be also interested in my PhD thesis arxiv.org/abs/1909.00744 (non-linear elliptic operators are discussed in section 2.2.4) $\endgroup$
    – Tobias Diez
    Commented Apr 18, 2020 at 12:43
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    $\begingroup$ Aren't there still two differences? 1) Hamilton's conclusions assert tameness, but nothing about the degree of tameness; 2) the conclusion on smoothness is as a map $C^\infty\to C^\infty$, which doesn't (?) automatically extend to smoothness as a map $C^{k,\alpha}\to C^{k+m,\alpha}$ $\endgroup$
    – Quarto Bendir
    Commented Apr 18, 2020 at 13:26
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    $\begingroup$ How do you establish the existence of $Q$? Did you use a Nash-Moser argument? Or probably something better, because normally the resulting $m$ would be negative and $k$ would have to be sufficiently. large. $\endgroup$
    – Deane Yang
    Commented Apr 18, 2020 at 18:59
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    $\begingroup$ In the elliptic case, you have a smooth map $F: A \rightarrow B$, where the linearization $F'(a): A \rightarrow B$ is is smooth and has a bounded inverse. In that case, I believe everything follows by the same argument you would use in the finite dimensional inverse function theorem. It basically follows by implicit differentiation. $\endgroup$
    – Deane Yang
    Commented Apr 19, 2020 at 1:27

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