Essential norm estimates for Hankel operators on convex domains in $\mathbb{C}^2$

Let $Ω\subset \mathbb{C}^2$ be a bounded convex domain with $C^1$-smooth boundary and $φ\in C^1(\overlineΩ)$ such that $φ$ is harmonic on the nontrivial disks in the boundary. We estimate the essential norm of the Hankel operator $H_φ$ in terms of the $\overline{\partial}$ derivatives of $φ$ "along" the nontrivial disks in the boundary.

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