On absolutely continuous weakly mixing cocycles over irrational rotations

A weakly mixing cocycle over a rotation $\alpha$ is a measurable function $\varphi\colon S^1\to S^1$, where $S^1=\{z\in\mathbb C:|z|=1\}$, such that the equation
\begin{equation} \varphi^n(z)=c\frac{h(\exp(2\pi i\alpha)z)}{h(z)} \quad\text{for almost all \ </nomathmode><mathmode>$z$} \tag{1} \end{equation}
</mathmode><nomathmode> has no measurable solutions $h(\,\cdot\,)\colon S^1\to S^1$ for any $n\in\mathbb Z\setminus\{0\}$ and $c\in\mathbb C$, $|c|=1$.
If the irrational number $\alpha$ has bounded convergents in its continued fraction expansion and a function $M(y)$ increases more slowly than $y\ln^{1/2}y$, then it is proved that there exists a weakly mixing cocycle of the form $\varphi(\exp(2\pi ix))=\exp(2\pi i\widetilde\varphi(x))$, where $\widetilde\varphi\colon\mathbb T\to\mathbb R$ belongs to the class $W^1(M(L)(\mathbb T))$. In addition, it is shown that equation (1) (and also the corresponding additive cohomological equation) is soluble for $\widetilde\varphi\in W^1(L\log_+^{1/2}L(\mathbb T))$.