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

Let $\mathrm{\Omega }\subset {\mathbb{C}}^2$ be a bounded convex domain with $C^1$-smooth boundary and $\phi \in C^1(\bar{\mathrm{\Omega }})$ such that $\phi$ is harmonic on the nontrivial disks in the boundary. We estimate the essential norm of the Hankel operator $H_{\phi }$ in terms of the $\bar{\mathrm{\partial }}$ derivatives of $\phi$ “along” the nontrivial disks in the boundary.