Decay estimates for the arithmetic means of coefficients connected with composition operators

Abstract

Let $f\in L^{\infty }(T)$ with $\Vert f\Vert _{\infty }\le 1$. If $f(0)\ne 0,$$n,k\in {\mathbb {Z}}$, and $b_{n,n-k}=\int _{E}f(x)^{n}e^{-2\pi i(n-k)x}dx$, $E=\{x\in T:|f(x)|=1\}$, we prove that the arithmetic means $\frac{1}{N} \sum _{n=M}^{M+N}|b_{n,n-k}|^{2}$ decay like $\{\log N\log _{2}N\cdot \cdot \cdot \log _{q}N\}^{-1}$ as $N\rightarrow \infty$, uniformly in $k\in {\mathbb {Z}}$.