The nonlocal Liouville-type equation in R and conformal immersions of the disk with boundary singularities

In this paper we perform a blow-up and quantization analysis of the fractional Liouville equation in dimension 1. More precisely, given a sequence $u_k : \mathbb{R} \to \mathbb{R}$ of solutions to
\begin{align}
(-\Delta)^{\frac{1}{2}})u_k = K_k e^{u_k} \text{ in } \mathbb{R},\tag{1}
\end{align}
with $K_k$ bounded in $L^{\infty}$ and $e^{u_k}$ bounded in $L^{1}$ uniformly with respect to $k$, we show that up to extracting a subsequence $u_k$ can blow-up at (at most) finitely many points $B=\{a_1,...,a_N\}$ and either (i) $u_k\to u_\infty$ in $W_{loc}^{1,p}(\mathbb{R}\setminus B)$ and $K_k e^{u_k} \stackrel{*}{\rightharpoonup} K_\infty e^{u_\infty} + \sum_k^N\pi\delta_{a_j}$ or (ii) $u_k\to u_\infty$ in $\mathbb{R}\setminus B$ and $K_k e^{u_k} \stackrel{*}{\rightharpoonup} \sum_k^N\alpha_j\delta_{a_j}$ with $\alpha_j \ge \pi$ for every $j$. This result, resting on the geometric interpretation and analysis of $(1)$ provided in a recent collaboration of the authors with T. Rivière and on a classical work of Blank about immersions of the disk into the plane, is a fractional counterpart of the celebrated works of Brézis-Merle and Li-Shafrir on the 2-dimensional Liouville equation, but providing sharp quantization estimates ($\alpha_j=\pi$ and $\alpha_j\ge\pi$) which are not known in dimension 2 under the weak assumption that $(K_k)$ be bounded in $L^\infty$ and is allowed to change sign.