OP-00316 Solution claimed

The sharp Beurling–Ahlfors bound for planar elliptic systems

For $f \in C_c^\infty(\mathbb{C}; \mathbb{C})$, with planar Lebesgue measure $dA$, let $$\mathcal{B} f(z) = -\frac{1}{\pi} \lim_{\varepsilon \downarrow 0} \int_{|z - w| > \varepsilon} \frac{f(w)}{(z - w)^2}\, dA(w)$$ be the Beurling–Ahlfors transform, and for $1 < p < \infty$ put $p^* = \max\{p, p/(p-1)\}$. On $L^2(\mathbb{C})$, $\mathcal{B}$ is an isometry. Lehto proved that the norm of $\mathcal{B}$ on $L^p(\mathbb{C})$ is at least $p^* - 1$, and Bañuelos and Janakiraman proved that it is at most $1.575\,(p^* - 1)$.

Problem. Is $$\|\mathcal{B} f\|_{L^p(\mathbb{C})} \le (p^* - 1)\, \|f\|_{L^p(\mathbb{C})}$$ for every $1 < p < \infty$ and every $f \in C_c^\infty(\mathbb{C}; \mathbb{C})$?

Source: Iwaniec (1982), who conjectured that the L^p norm of the Beurling–Ahlfors transform is p* − 1; the lower bound is Lehto's (1965).

Imported from github.com/MColbrook/AIM.

Classification

Primary: CA — Classical Analysis and ODEs
Secondary: AP, CV
Keywords: AIM Open Applied Problems, Beurling–Ahlfors transform, quasiconformal mappings, singular integrals

References

@article{iwaniec1982,
  author  = {Iwaniec, Tadeusz},
  title   = {Extremal inequalities in {S}obolev spaces and quasiconformal mappings},
  journal = {Z. Anal. Anwendungen},
  volume  = {1},
  number  = {6},
  year    = {1982},
  pages   = {1--16},
  doi     = {10.4171/ZAA/37},
}
@article{lehto1965,
  author  = {Lehto, Olli},
  title   = {Remarks on the integrability of the derivatives of quasiconformal mappings},
  journal = {Ann. Acad. Sci. Fenn. Ser. A I},
  number  = {371},
  year    = {1965},
  note    = {8 pp.},
}
@article{banuelos2008,
  author  = {Ba{\~n}uelos, Rodrigo and Janakiraman, Prabhu},
  title   = {{$L^p$}-bounds for the {B}eurling--{A}hlfors transform},
  journal = {Trans. Amer. Math. Soc.},
  volume  = {360},
  number  = {7},
  year    = {2008},
  pages   = {3603--3612},
  doi     = {10.1090/S0002-9947-08-04537-6},
}
@misc{aim-beurling-ahlfors-sharp-norm,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{The sharp Beurling–Ahlfors bound for planar elliptic systems}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/446-beurling-ahlfors-sharp-norm.md},
  note         = {Entry 446 at commit a153f46, 2026-10-13},
}

Resolution claims

Sharp Higher Integrability Theory, Part I: The $L^p$-norm of the Beurling--Ahlfors transform v1 arXiv ↗ AI-Verified

Andrea Agazzi; Kari Astala; Rodrigo Bañuelos; Giuseppe Bruno; Gabriele Cassese; Daniel Faraco; André Guerra; Mahdi Hormozi; Bernd Kirchheim; Aleksis Koski; Vjekoslav Kovač; Jan Kristensen; Federico Pasqualotto; István Prause; Eero Saksman; Vladimír Šverák; László Székelyhidi; Riccardo Tione; Alexander Volberg; Hang Yang

We confirm a conjecture posed by T. Iwaniec in 1982: the $L^p$ norm of the Beurling--Ahlfors transform is $p-1$ if $p\geq 2$, and $(p-1)^{-1}$ if $1<p\leq 2$.

Full resolution · not verified · fidelity not established · posted on arXiv 8 Oct 2026

Status history

When	Change	By	Reason
2026-10-15	Needs verification → Solution claimed	—	Recorded here: arXiv:2609.28311v2.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00316, "The sharp Beurling–Ahlfors bound for planar elliptic systems", clopen.solutions.