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

The 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]

What gets accepted

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