The critical-index boundary conjecture for Shimorin-type operators

Let $\nu$ be a finite positive Borel measure on $[0,1]$, and let $T_\nu:L^p(\mathbb{D})\to L^q(\mathbb{D})$ be the associated Shimorin-type operator. Define

$$c_\nu=\sup\left\{1\leq c<2:\int_0^1\frac{1}{(1-r^2)^{2/c'}}\,d\nu(r)<\infty\right\},$$

where $c'$ is the dual index, satisfying $1/c+1/c'=1$. The critical-index boundary conjecture. The boundary line $\mathcal{C}$ between the $L^p$-$L^q$ boundedness and unboundedness regions of $T_\nu$ is given by

$$\mathcal{C}=\left\{\left(\frac1p,\frac1q\right):1\leq p\leq c_\nu',\quad \frac1q=\frac1p+\frac1{c_\nu}-1\right\}.$$

This conjecture proposes that the quantity $c_\nu$ plays for Shimorin-type operators the role played by the singularity parameter for Bergman-type operators. Its validity is not established in the supplied text and remains open.