Beurling–Ahlfors transform constant

Description of constant

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]

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

Contribution notes

Prepared with assistance from ChatGPT 5.2 Pro.