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

Abstract

The operators ${\Lambda }_m$ ($m\in \mathbb{N}\cup \{0\}$) 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 ${\Lambda }_0$ is equal to $1/\ln (2)\approx 1.44$. We construct examples showing that the weak-type $(1,1)$ constant of ${\Lambda }_1$ is larger than $1.38$ and that the weak-type $(1,1)$ constant of ${\Lambda }_m$ does not tend to $1$ when $m\to \infty$. 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.

50 pages, 10 figures, 1 table (includes four long appendices); presentation streamlined: the main body of the article is now 16 pages, while the most detailed and technical parts of the article were moved to the appendix

References in corpus (1)

• Hardy's operator minus identity and power weights