Heating of the Ahlfors–Beurling operator, and estimates of its norm

Volberg, A.; Nazarov, F.

doi:10.1090/S1061-0022-04-00822-2

Heating of the Ahlfors–Beurling operator, and estimates of its norm
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by A. Volberg and F. Nazarov
Translated by: the authors

St. Petersburg Math. J. 15 (2004), 563-573

DOI: https://doi.org/10.1090/S1061-0022-04-00822-2

Published electronically: July 6, 2004

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Abstract:

A new estimate is established for the norm of the Ahlfors–Beurling transform $T\varphi (z):=\frac 1\pi \iint \frac {\varphi (\zeta ) dA(\zeta )}{(\zeta - z)^2}$ in $L^p(dA)$. Namely, it is proved that $\|T\|_{L^p\rightarrow L^p} \leq 2(p-1)$ for all $p\geq 2$. The method of Bellman function is used; however, the exact Bellman function of the problem has not been found. Instead, a certain approximation to the Bellman function is employed, which leads to the factor 2 on the right (in place of the conjectural $1$).

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