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Hans Lewy shows in his 1957 example that a certain linear PDE with polynomial coefficients cannot have a solution with Hölder derivatives in any open subset of $\mathbb{R}^3$.

I find this example cited everywhere as a "linear equation without solution"; however that is not what appears to be shown in the paper, since (if I recall correctly) there are $C^1$ functions that are nowhere Hölder.

Is a (hypothetical) "solution" with non-Hölder derivatives simply not considered "well-behaved" enough to be called a solution? Are there other results related to that paper that go further?

What is the point I am missing here?

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Indeed, the introduction of Hölder's condition in the second part of Lewy's paper is a bit unsatisfactory. Hartman removed it 1-2 years later (depending on if you look at the publication or submission dates). His paper is in open access (three cheers to the AMS).

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