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a) How to construct a conformal mapping $f$ of the unit disk $D$ onto a Jordan domain with $C^1$ boundary such that $$\int_D|f''(z)|^2 dxdy =\infty.$$ (This is done in two different ways in the sequel)

b) It follows by Kellogg theorem that if $\partial D\in C^{1,\alpha}$ with $\alpha>1/2$, then $\int_D|f''(z)|^2 dxdy <\infty$

c) Motivated by b), I expect that if $\partial D\in C^{1,\alpha}$ with $0<\alpha\le 1/2$, then $\int_D|f''(z)|^2 dxdy <\infty$.

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  • $\begingroup$ Is the Jordan domain fixed or do you want to construct both the mapping and its image? Two conformal bijections $f,g:D\to U\subset\mathbb C$ only differ by a Möbius automorphism $\phi$ of $D$: we have $f=g\circ\phi$. Since $\phi\in C^2(\bar D,\bar D)$, $f'\in L^2$ and $\phi'$ is bounded away from zero, the infinitude of your integral is independent of $\phi$. $\endgroup$
    – Joonas Ilmavirta
    Commented Sep 18, 2014 at 18:18
  • $\begingroup$ @user57714: I think that if you want more people to think about your question c), you should ask it as a separate question. This being said, a result again from Pommerenke's book implies that if the boundary of $\Omega$ is $C^{1,\alpha}$ for some $0<\alpha<1$, then the Riemann map $f:\mathbb{D} \to \Omega$ extends to a function $f \in C^1(\overline{\mathbb{D}})$. This, however, does not imply that the integral you wrote is finite... $\endgroup$
    – Malik Younsi
    Commented Sep 19, 2014 at 23:20

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Let $g$ be analytic in $\mathbb{D}$, continuous on $\overline{\mathbb{D}}$ such that $\int \int_{\mathbb{D}} |g'|^2=\infty$. It is well-known that such functions exist (the disc algebra is not contained in the Dirichlet space).

We can assume that $\operatorname{Re} g>0$ and $ | \operatorname{Im} g |<\pi/2$ on $\mathbb{D}$.

Then let $f$ such that $f'=e^g$. Since $|\arg{f'}|=|\operatorname{Im}g|<\pi/2$, we have that $\operatorname{Re}f'>0$ on $\mathbb{D}$, which is enough to conclude that $f$ maps $\mathbb{D}$ conformally onto some Jordan domain $G$ (see e.g. Pommerenke's book Boundary behaviour of conformal maps, Proposition 1.1).

Also, $\arg{f'}=\operatorname{Im}g$ is continuous on $\overline{\mathbb{D}}$, and so $\partial G$ is $C^1$ by Theorem 3.2 in Pommerenke's book.

Finally, $\int \int_{\mathbb{D}} |f''|^2=\int \int_{\mathbb{D}}|g'e^g|^2 \geq \int \int_{\mathbb{D}}|g'|^2 = \infty,$ since $|e^g|=e^{\operatorname{Re}g}>1$.

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You should take a look at an example in a paper by Lesley and Warschawski published in Math Z.

The article is called "On conformal mappings with Derivative in VMOA" and is available at https://eudml.org/doc/172637.

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    $\begingroup$ Please provide full details when you have an opportunity. $\endgroup$
    – Todd Trimble
    Commented Sep 18, 2014 at 19:38
  • $\begingroup$ Glancing over the paper, the final part of their Theorem states that there exist domains with smooth boundary so that the derivative of the mapping function does not have Bounded Mean Oscillation. This does not seem to formally imply an answer to the question (?), but may be a good place to start. $\endgroup$
    – Lasse Rempe
    Commented Sep 19, 2014 at 10:59
  • $\begingroup$ Another place to look for results and references may Pommerenke's "Boundary Behaviour of conformal maps". I don't have it to hand at the moment. $\endgroup$
    – Lasse Rempe
    Commented Sep 19, 2014 at 11:00
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The function $f(z)= z (\pi + 2 i \log z)$ maps a Jordan domain $\Omega$ bounded by a $C^2$ curve containing $(-1/2,1/2)$ onto a domain $\Omega'$ with $C^1$ boundary. Then $f''(z)=2/z$ so the given integral diverges. It can be composed by a conformal mapping of the unit disk onto $\Omega$ such that the given integral diverges as well.

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