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Let $D$ be a Jordan domain. We assume that $D$ is a Hölder domain. Namely, there exists a Hölder continuous Riemann map $f$ such that $D=f(\mathbb{D})$, where $\mathbb{D}$ is an open unit disk.

It is known that quasidisks are Hölder domains. Hence, the boundaries of Hölder domains can be very wild. For example, the Koch snowflake is a Hölder domain.

My question

How strange is $D$?

I would like to know an example of Jordan domain which is Hölder and does not possesses both volume doubling property and the uniform poincare inequality. These two notions are fundamental in geometric analysis.

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  • $\begingroup$ See mathoverflow.net/questions/336224/… for the definition of the uniform poincare inequality. $\endgroup$
    – sharpe
    Commented Jul 27, 2019 at 15:46
  • $\begingroup$ Thank you for your comment, I should edit as you say. $\endgroup$
    – sharpe
    Commented Jul 27, 2019 at 19:58
  • $\begingroup$ while you are editing you may also check the spelling of Hölder :) $\endgroup$
    – Pietro Majer
    Commented Jul 27, 2019 at 20:33
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    $\begingroup$ @PietroMajer Thank you for your comment :) $\endgroup$
    – sharpe
    Commented Jul 28, 2019 at 5:23
  • $\begingroup$ "Hölder older"... What is this... $\endgroup$
    – sharpe
    Commented Aug 4, 2019 at 5:31

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