Lower bounds for the weak-type constants of the operators ${\mathrm{\Lambda }}_m$

The operators ${\mathrm{\Lambda }}_m$ ($m\in \mathbb{N}\cup \{0\}$) arise when one studies the action of the Beurling-Ahlfors transform on certain radial function subspaces. It is known that the weak-type $(1,1)$ constant of ${\mathrm{\Lambda }}_0$ is equal to $1\mathrm{/}\ln (2)\approx 1.44$. We construct examples showing that the weak-type $(1,1)$ constant of ${\mathrm{\Lambda }}_1$ is larger than $1.38$ and that the weak-type $(1,1)$ constant of ${\mathrm{\Lambda }}_m$ does not tend to $1$ when $m\to \mathrm{\infty }$. This disproves a conjecture of Gill [Mich. Math. J. 59 (2010), No. 2, 353-363]. We also prove a companion result for the adjoint operators. This is the arXiv version of the paper - it includes some additional discussion in the appendices.