Carleson measures induced by higher order Schwarzian derivatives and derivatives of analytic functions

Abstract

Let G be a finitely generated Fuchsian group of the second kind without any parabolic element and f be a univalent analytic function in the unit disk D compatible with G. In this paper, we study the higher order Schwarzian derivatives: \sigma_{n+1}(f)=\sigma'_{n}(f)-(n-1)\frac{f''}{f'}\cdot\sigma_{n}(f),  n\geq3, where \sigma_{3}(f) stands for the Schwarzian derivatives of f, and 
S_{n}(f)=(f')^{\frac{n-1}{2}}D^{n}(f')^{-\frac{n-1}{2}},n\geq2.
For p>0, we show that if |\sigma_{n}(f)(z)|^{p}(1-|z|^{2})^{p(n-1)-1}dxdy (resp. |S_{n}(f)(z)|^{p}(1-|z|^{2})^{pn-1}dxdy) satisfies the Carleson condition on the infinite boundary of the Dirichlet fundamental domain $\mathcal{\mathcal{F}}$ of $G$, then $|\sigma_{n}(f)(z)|^{p}(1-|z|^{2})^{p(n-1)-1}dxdy$ (resp. $|S_{n}(f)(z)|^{p}(1-|z|^{2})^{pn-1}dxdy$) is a Carleson measure in $\mathbb{D}$. Similarly, for $p>0$ and a bounded analytic function $f$ in the unit disk $\mathbb{D}$ compatible with $G$, we prove that if $|f'(z)|^{p}(1-|z|^{2})^{p-1}dxdy$ satisfies the Carleson condition on the infinite boundary of the Dirichlet fundamental domain $\mathcal{\mathcal{F}}$ of $G$, then $|f'(z)|^{p}(1-|z|^{2})^{p-1}dxdy$ is a Carleson measure in $\mathbb{D}$.