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From the Anderson-Kadec theorem, we know that all separable infinite-dimensional Banach spaces are homeomorphic. I'm wondering, is there an explicit such homeomorphism between $W^{p,k}(\mathbb{R}^n)$ and $L^2(\mu,\mathbb{R}^n)$? Here, $\mu$ is a Borel probability measure on $\mathbb{R}^n$ and $W^{p,k}(\mathbb{R}^n)$ is the Sobolev space of "functions" with $k$ weak derivates with finite $p^{th}$ norm. If possible, an explicit bi-Lipschitz homeomorphism would be most interesting.

I know that the Fourier transform does the job when $\mu$ is the Lebesgue measure and $k$ is sufficiently well-chosen, but I was wondering what is possible in this setting.

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    $\begingroup$ You already know that the Sobolev space and $L_p(\lambda)$ are explicily linearly isomorphic; also $L_2(\mu)$ and $L_2(\lambda)$ are isometric (via orthonormal bases, hence (?) explicitly); finally $L_p(\mu)$ and $L_2(\mu)$ are homeomorphic via the Mazur map $f\mapsto |f|^{p/2}\operatorname{sign}(f)$. (Mazur's paper is in Studia Math.\ 1, 83--85 (1929).) $\endgroup$
    – Dirk Werner
    Commented Jun 9, 2020 at 15:53
  • $\begingroup$ @DirkWerner Yes but the Isomorphism between the Sobolev space and $L_p(\lambda)$ is practical like the Mazur map in the sense that a function must first be decomposed in terms of an basis and then mapped. Right? Is there a direct transformation does the job (even if it's non-linear like the Mazur map)? $\endgroup$
    – ABIM
    Commented Jun 11, 2020 at 6:53
  • $\begingroup$ @AIM_BPB For the isomorphism between the Sobolev space and $L_p$ one uses Fourier multipliers, right? So only the isomorphism between the $L_2$-spaces requires the choice of a noncanonical basis. $\endgroup$
    – Dirk Werner
    Commented Jun 11, 2020 at 18:47
  • $\begingroup$ Actually I'm not familiar with Fourier multipliers. (now this may have become basic) but how would you do that? $\endgroup$
    – ABIM
    Commented Jun 12, 2020 at 11:24
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    $\begingroup$ Please take a look at Wojtaszczyk's ``Banach Spaces For Analysts'', Proposition III.A.3 (which deals with the periodic Sobolev space). $\endgroup$
    – Dirk Werner
    Commented Jun 12, 2020 at 17:28

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