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Heating of the Ahlfors-Beurling operator, and estimates of its norm. (English. Russian original) Zbl 1061.47042

St. Petersbg. Math. J. 15, No. 4, 563-573 (2004); translation from Algebra Anal. 15, No. 4, 142-158 (2003).
Authors’ abstract: A new estimate is established for the norm of the Ahlfors-Beurling transform \(T\phi(z):=\frac{1}{\pi}\iint\frac{\phi(\zeta) dA(\zeta)}{(\zeta-z)^2}\) in \(L^{p}(dA).\) Namely, it is proved that \(\| T\| _{L^{p}\to L^{p}}\leq 2(p-1)\) 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 \(1\)).

MSC:

47G10 Integral operators
45P05 Integral operators
35K05 Heat equation
47B38 Linear operators on function spaces (general)
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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