Iwaniec’s Beurling–Ahlfors norm conjecture

Background and motivation

The conjecture concerns a singular integral operator important in both partial differential equations and quasiconformal mapping theory. The 2026 preprint describes the conjecture as central to geometric function theory and explains that the sharp norm bound yields sharp higher integrability for derivatives of planar quasiconformal maps. It presents its proof as part of a broader higher-integrability program. (Agazzi et al., Introduction)

History

A historical account by Rodrigo Bañuelos and Prabhu Janakiraman attributes the lower bound $\|B\|_p\ge p^*-1$ to O. Lehto and identifies Iwaniec’s proposed equality as a conjecture. Their account reviews successive upper bounds: first $4(p^*-1)$, then $2(p^*-1)$, followed by their own bound $1.575(p^*-1)$. These are historical estimates reported in that paper, not claims about the best bound today. (Bañuelos and Janakiraman, PDF pp. 1–2) The paper’s references identify Iwaniec’s cited conjecture source as a 1982 publication. (Bañuelos and Janakiraman, PDF p. 11)

Known results and status

The preliminary preprint Sharp Higher Integrability Theory, Part I claims the exact norm formula in its Corollary 2. Its introduction says the lower bound is classical and identifies the upper bound as the new contribution; it also states that the corollary settles Iwaniec’s conjecture. This is a claim made in the September 2026 preprint, rather than an independently established result in the supplied sources. (Agazzi et al., Introduction and Corollary 2)

• Sharp Higher Integrability Theory, Part I: The Lp-norm of the Beurling--Ahlfors transform — arXiv — Andrea Agazzi, Kari Astala, Giuseppe Bruno, Gabriele Cassese, Hang Chang, Daniel Faraco, André Guerra, Bernd Kirchheim, Aleksis Koski, Jan Kristensen, Federico Pasqualotto, István Prause, Riccardo Tione, Vladimír Šverák, László Székelyhidi