Weighted estimates for powers of the Ahlfors-Beurling operator

Dragičević, Oliver

doi:10.1090/S0002-9939-2010-10645-7

Weighted estimates for powers of the Ahlfors-Beurling operator
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Proc. Amer. Math. Soc. 139 (2011), 2113-2120 Request permission

Abstract:

We prove that for any $n\in \mathbb {Z}\backslash \{0\}$, $p>1$ and any weight $w$ from the Muckenhoupt $A_p$ class, the norm of the $n$-th power of the Ahlfors-Beurling operator $T$ on the weighted Lebesgue space $L^p(w)$ is majorized by $C(p) |n|^3 [w]_p^{\operatorname {max}\{1,1/(p-1)\}}$, where $[w]_p$ is the $A_p$ characteristic of $w$. We apply this estimate for a result concerning the spectrum of $T$ on $L^p(w)$.

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