In the similar context as Conformal welding of rectifiable curves

In classical conformal welding theory, we start with a homeomorphism $h$ of the unit circle and try to find a Jordan domain $D$ together with two conformal isomorphisms $f_1 \colon \mathbb D \to D$ and $f_2 \colon \hat{\mathbb C} \backslash \overline{\mathbb D} \to \hat{\mathbb C} \backslash \overline{D}$. such that $h|_{S^1} = f_1^{-1} \circ f_2$. The existence of these two maps are guaranteed if $h$ is a quasisymmetric homeomorphism, and the uniqueness follows from uniqueness of complex dilatation.

I showed existence of conformal welding for a particular non-quasisymmetric map $h$ using a condition called Lehto condition (see "Random Conformal Weldings" for the setting).

This resulted in the above mentioned conformal maps and a Jordan loop $\Gamma=\partial D$.

What can we try to prove about $\Gamma$ using the particular properties of $h$? For example, can we prove any multifractal spectrum questions about $\Gamma$?

I understand this question is a bit open-ended but I am not familiar with the literature. Other than that there a unique corresponded in the quasisymmetric setting, I don't know whether the quasisymmetric maps can be used to be compute properties about the Jordan loop $\Gamma$.

For example, I don't know if the answer is trivially no "no very little can be inferred" or trivially yes "yes many properties can be inferred eg. see work on...".

Bishop in the work "Conformal welding and Koebe's theorem", manages to build some approximations to the loop $\Gamma$ but I am not sure what information is preserved in the limit.

In the similar context as Conformal welding of rectifiable curves

In classical conformal welding theory, we start with a homeomorphism $h$ of the unit circle and try to find a Jordan domain $D$ together with two conformal isomorphisms $f_1 \colon \mathbb D \to D$ and $f_2 \colon \hat{\mathbb C} \backslash \overline{\mathbb D} \to \hat{\mathbb C} \backslash \overline{D}$. such that $h|_{S^1} = f_1^{-1} \circ f_2$. The existence of these two maps are guaranteed if $h$ is a quasisymmetric homeomorphism, and the uniqueness follows from uniqueness of complex dilatation.

I showed existence of conformal welding for a particular non-quasisymmetric map $h$ using a condition called Lehto condition (see "Random Conformal Weldings" for the setting).

This resulted in the above mentioned conformal maps and a Jordan loop $\Gamma=\partial D$.

What can we try to prove about $\Gamma$ using the particular properties of $h$? For example, can we prove any multifractal spectrum questions about $\Gamma$?

I understand this question is a bit open-ended but I am not familiar with the literature. Other than that there is unique correspondence in the quasisymmetric setting, I don't know whether the quasisymmetric welding maps can be used to be compute properties about the resulting Jordan loop $\Gamma$.

For example, I don't know if the answer is trivially no "no very little can be inferred" or trivially yes "yes many properties can be inferred eg. see work on...".

Bishop in the work "Conformal welding and Koebe's theorem", manages to build some approximations to the loop $\Gamma$ but I am not sure what information is preserved in the limit.