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 $\int\_D|f''(z)|^2 dxdy <\infty$. 



Integrability of second derivative of conformal mappings
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 $|e^g|=e^{\operatorname{Re}g}>1$.


