Weighted estimates for powers of the Ahlfors-Beurling operator

Abstract

We prove that for any n ∈ Z\{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] max{1,1/(p―1)} p , 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).