OP-12066OpenAnalysis

Beurling–Ahlfors transform constant ( $C_{54}$ )

Beurling–Ahlfors transform constant: determine $C_{54}$ or improve its bounds; the repository records 1 (lower) and 1.575 (upper).

MSC 42B20 · Singular and oscillatory integrals (Calderón-Zygmund, etc.)MSC 30C62 · Quasiconformal mappings in the complex plane

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓

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Problem statement

In harmonic analysis, the Beurling–Ahlfors transform $B$ (also called the Ahlfors–Beurling operator) is the singular integral operator on $L^p(\mathbb{C})$ , $1<p<\infty$ , defined by

Bf(z)\ =\ -\frac{1}{\pi}\,\operatorname{p.v.}\int_{\mathbb{C}} \frac{f(w)}{(z-w)^2}\,dm(w) \ =\ -\frac{1}{\pi}\,\lim_{\varepsilon\to 0^+}\int_{\lvert w-z\rvert>\varepsilon}\frac{f(w)}{(z-w)^2}\,dm(w),

where $dm$ is Lebesgue measure on $\mathbb{C}$ . [BJ2008-def-B]

For $1<p<\infty$ , set

p^*\ :=\ \max\left\{p,\frac{p}{p-1}\right\}.

[BJ2008-abs-conj-pstar]

Write $\lVert B\rVert_p := \lVert B\rVert_{L^p(\mathbb{C})\to L^p(\mathbb{C})}$ .

We define

C_{54}\ :=\ \sup_{1<p<\infty}\ \frac{\lVert B\rVert_p}{p^*-1},

equivalently, the smallest constant $C$ such that $\lVert B\rVert_p \le C\,(p^*-1)$ for all $1<p<\infty$ .

The operator $B$ is an isometry on $L^2(\mathbb{C})$ , hence $\lVert B\rVert_2=1$ . [BJ2008-L2-isometry]

Determining $\lVert B\rVert_p$ for $1<p<\infty$ is an outstanding open problem. [BJ2008-open-problem]

Lehto proved the lower bound $\lVert B\rVert_p \ge p^*-1$ , and Iwaniec conjectured that equality holds for all $1<p<\infty$ . [BJ2008-Lehto-lb] [BJ2008-Iwaniec-conj]

The best currently proved uniform upper estimate is $\lVert B\rVert_p \le 1.575\,(p^*-1)$ , so the rigorous range is

1\ \le\ C_{54}\ \le\ 1.575.

[BJ2008-abs-ub1575] [BJ2008-Lehto-lb]

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Background

Conjectural value. Iwaniec conjectured (in [Iwa1982]) that $\lVert B\rVert_p = p^*-1$ for all $1<p<\infty$ , i.e., $C_{54}=1$ . [BJ2008-Iwaniec-conj] [BJ2008-ref-Iwa1982]
Wikipedia page on the Beurling transform

References (from the repository page)

[BJ2008] Bañuelos, Rodrigo; Janakiraman, Prabhu. Lp-bounds for the Beurling–Ahlfors transform. Transactions of the American Mathematical Society 360 (2008), no. 7, 3603–3613. PDF: http://www.ams.org/tran/2008-360-07/S0002-9947-08-04537-6/S0002-9947-08-04537-6.pdf. Google Scholar
[BJ2008-abs-conj-pstar] loc: BJ2008 PDF p.1 (Abstract) quote: “The celebrated conjecture of T. Iwaniec states that its $L^p$ norm $\lVert B\rVert_p = p^* - 1$ where $p^*=\max\{p,\frac{p}{p-1}\}$ .”
[BJ2008-abs-ub1575] loc: BJ2008 PDF p.1 (Abstract) quote: “the new upper estimate $\lVert B\rVert_p \le 1.575(p^* - 1),\ 1<p<\infty$ is found.”
[BJ2008-def-B] loc: BJ2008 PDF p.1 (Section 1: Introduction), Eq. (1.1) quote: “This singular integral operator $B$ defined on $L^p(\mathbb{C})$ , $1<p<\infty$ , by $Bf(z) = -\frac{1}{\pi}\,\mathrm{p.v.}\int_{\mathbb{C}} \frac{f(w)}{(z-w)^2}\,dm(w)$ .”
[BJ2008-L2-isometry] loc: BJ2008 PDF p.1 (Section 1: Introduction) quote: “Thus, $B$ is an isometry on $L^2(\mathbb{C})$ and in particular $\lVert B\rVert_2 = 1$ .”
[BJ2008-open-problem] loc: BJ2008 PDF p.1 (Section 1: Introduction) quote: “An outstanding open problem of the past 25 years is the computation of its $L^p$ norm for $1<p<\infty$ .”
[BJ2008-Lehto-lb] loc: BJ2008 PDF p.1 (Section 1: Introduction) quote: “In [16], Lehto shows that $\lVert B\rVert_p \ge p^* - 1$ .”
[BJ2008-Iwaniec-conj] loc: BJ2008 PDF p.1 (Section 1: Introduction) quote: “T. Iwaniec conjectures in [13] that $\lVert B\rVert_p = p^* - 1$ .”
[BJ2008-ub4] loc: BJ2008 PDF p.1 (Section 1: Introduction) quote: “Bañuelos and Wang use the martingale inequalities of Burkholder to prove the preliminary upper bound $\lVert B\rVert_p \le 4(p^* - 1)$ .”
[BJ2008-ub2] loc: BJ2008 PDF p.2 (Section 1: Introduction) quote: “Nazarov and Volberg [17] lower the bound to $2(p^* - 1)$ .”
[BJ2008-ref-Iwa1982] loc: BJ2008 PDF p.11 (References) quote: “[13] T. Iwaniec; Extremal inequalities in Sobolev spaces and quasiconformal mappings. Z. Anal. Anwendungen 1 (1982), 1–16.”
[BW1995] Bañuelos, Rodrigo; Wang, Gang. Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms. Duke Mathematical Journal 80 (1995), no. 3, 575–600. DOI: https://doi.org/10.1215/S0012-7094-95-08020-X. Google Scholar
[NV2004] Nazarov, Fedor; Volberg, Alexander. Heat extension of the Beurling operator and estimates for its norm. St. Petersburg Mathematical Journal 15 (2004), no. 4, 563–573. DOI: https://doi.org/10.1090/S1061-0022-04-00822-2. Google Scholar
[Leh1965] Lehto, Olli. Remarks on the integrability of the derivatives of quasiconformal mappings. Annales Academiae Scientiarum Fennicae. Series A I. Mathematica 371 (1965), 8 pp. PDF: https://www.acadsci.fi/mathematica/1966/no371pp01-08.pdf. Google Scholar
[Iwa1982] Iwaniec, Tadeusz. Extremal inequalities in Sobolev spaces and quasiconformal mappings. Zeitschrift für Analysis und ihre Anwendungen 1 (1982), no. 6, 1–16. DOI: https://doi.org/10.4171/ZAA/37. Google Scholar

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 1; best known upper bound 1.575. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound	Reference	Comments
$4$	[BW1995]	Bañuelos–Wang proved $\lVert B\rVert_p \le 4(p^*-1)$ . [BJ2008-ub4]
$2$	[NV2004]	Nazarov–Volberg improved this to $\lVert B\rVert_p \le 2(p^*-1)$ . [BJ2008-ub2]
$1.575$	[BJ2008]	Bañuelos–Janakiraman proved $\lVert B\rVert_p \le 1.575(p^*-1)$ for all $1<p<\infty$ . [BJ2008-abs-ub1575]

Known lower bounds

Bound	Reference	Comments
$1$	[Leh1965]	Lehto proved $\lVert B\rVert_p \ge p^*-1$ , which implies $C_{54}\ge 1$ . [BJ2008-Lehto-lb]

References

[1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Beurling–Ahlfors transform constant ($C_{54}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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Source & evidence

https://teorth.github.io/optimizationproblems/constants/54a.html

Constant page from github.com/teorth/optimizationproblems at commit 2c1968c, Apache-2.0. The maintainers review submissions for plausibility and replicability but do not certify correctness; asterisked bounds have minimal verification.

Source date: Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differ
Imported: 2026-10-10

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Record

Difficulty: HardResisted sustained effort by specialists.
Posed by: Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers) and contributors
Posed: 2026
Source: Curated problem listOptimization Constants in Mathematics — Davis, Ivanisvili & TaoOptimization Constants in Mathematics — Davis, Ivanisvili & TaoOriginal statement ↗
Citation: Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Beurling–Ahlfors transform constant ($C_{54}$)
Added: 5 Nov 2026
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Analysis