𝐿^{𝑝}–bounds for the Beurling–Ahlfors transform

Bañuelos, Rodrigo; Janakiraman, Prabhu

doi:10.1090/S0002-9947-08-04537-6

$L^p$–bounds for the Beurling–Ahlfors transform
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by Rodrigo Bañuelos and Prabhu Janakiraman PDF

Trans. Amer. Math. Soc. 360 (2008), 3603-3612 Request permission

Abstract:

Let $B$ denote the Beurling–Ahlfors transform defined on $L^p(\mathbb {C})$, $1<p<\infty$. The celebrated conjecture of T. Iwaniec states that its $L^p$ norm $\|B\|_p=p^*-1$ where $p^*= \max \{p,\frac {p}{p-1}\}$. In this paper the new upper estimate \[ \|B\|_p\leq 1.575 (p^*-1), \hspace {3mm} 1<p<\infty ,\] is found.

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