Boundedness conjecture for wavelet-type Malmquist–Takenaka operators with bounded hyperbolic gaps

Let $D>0$ be a constant, and let $\mathbf{a}=(a_n)_{n=1}^{\infty}$ be a sequence such that the hyperbolic distance between $a_n$ and $a_{n+1}$ is at most $D$ for every $n=1,2,\dots$. Let $T^{\mathbf{a}}$ denote the associated wavelet-type Malmquist–Takenaka operator on $L^2(\mathbb{T})$. Boundedness conjecture. One has

$$\|T^{\mathbf{a}}\|_{L^2(\mathbb{T})\to L^2(\mathbb{T})}\lesssim_D 1.$$

This conjecture generalizes the paper's main boundedness results and is motivated by the conformal invariance of the problem and the expected cancellation produced by uniformly bounded hyperbolic gaps. Its resolution is not established in the supplied text.