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Subordination by conformal martingales in \(L^{p}\) and zeros of Laguerre polynomials. (English) Zbl 1266.32006
Summary: Given martingales \(W\) and \(Z\) such that \(W\) is differentially subordinate to \(Z\), D. Burkholder obtained the sharp inequality \(\mathbb E|W|^{p}\leq(p^{*}-1)^{p}\mathbb E|Z|^{p}\), where \[ p^{*}=\max\Big\{p,\frac{p}{p-1}\Big\}. \] What happens if one of the martingales is also a conformal martingale? R. Bañuelos and P. Janakiraman [Trans. Am. Math. Soc. 360, No. 7, 3603–3612 (2008; Zbl 1220.42012)] proved that if \(p\geq2\) and \(W\) is a conformal martingale differentially subordinate to any martingale \(Z\), then \[ \mathbb E|W|^{p}\leq \bigg [\frac{p^{2}-p}{2}\bigg]^{p/2}\mathbb E|Z|^{p}. \]
In this paper, we establish that if \(p\geq2\), \(Z\) is conformal, and \(W\) is any martingale subordinate to \(Z\), then \[ \mathbb E|W|^{p}\leq\bigg[\sqrt{2}\frac{1-z_{p}}{z_{p}}\bigg]^{p}\mathbb E|Z|^{p}, \] where \(z_{p}\) is the smallest positive zero of a certain solution of the Laguerre ordinary differential equation. We also prove the sharpness of this estimate and an analogous one in the dual case for \(1< p < 2\). Finally, we give an application of our results. Previous estimates on the \(L^{p}\)-norm of the Beurling-Ahlfors transform give at best \(\|B\|_{p}\lesssim\sqrt{2}p\) as \(p\rightarrow\infty\). We improve this to \(\|B\|_{p}\lesssim1.3922\,p\) as \(p\rightarrow\infty\).

MSC:
32A55 Singular integrals of functions in several complex variables
60G46 Martingales and classical analysis
42A15 Trigonometric interpolation
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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