Remark on Integral Means of Derivatives of Blaschke Products

Reijonen, Atte

Canadian mathematical bulletin, Tome 61 (2018) no. 3, pp. 640-649

Résumé

If $B$ is the Blachke product with zeros $\{{{z}_{n}}\},\,\text{then}\,\left| {B}'\left( z \right) \right|\,\le \,{{\Psi }_{B}}\left( z \right)$ , where $${{\Psi }_{B}}\,=\,\sum\limits_{n}{\frac{1-{{\left| {{z}_{n}} \right|}^{2}}}{{{\left| 1-{{\overline{z}}_{n}}z \right|}^{2}}}.}$$ Moreover, it is a well-known fact that, for $0\,<\,p\,<\,\infty $ , $${{M}_{p\left( r,{B}' \right)}}\,=\,{{\left( \frac{1}{2\pi }\,\int_{0}^{2\pi }{{{\left| {B}'\left( \text{r}{{\text{e}}^{i\theta }} \right) \right|}^{p}}d\theta } \right)}^{{1}/{p}\;}},\,0\,\le \,r\,<\,1,$$ is bounded if and only if ${{M}_{p}}\left( r,\,{{\Psi }_{B}} \right)$ is bounded. We find a Blaschke product ${{B}_{0}}$ such that ${{M}_{p}}\left( r,\,{{{{B}'}}_{0}} \right)$ and ${{M}_{p}}\left( r,{{\Psi }_{{{B}_{0}}}} \right)$ are not comparable for any $\frac{1}{2}\,<\,p\,<\,\infty $ . In addition, it is shown that, if $0\,<\,p\,<\,\infty$ , $B$ is a Carleson-Newman Blaschke product and a weight $\omega $ satisfies a certain regularity condition, then $${{\int }_{\mathbb{D}}}{{\left| {B}'\left( z \right) \right|}^{p}}\omega \left( z \right)dA\left( z \right)~\asymp {{\int }_{\mathbb{D}}}{{\Psi }_{B}}{{\left( z \right)}^{p}}\omega \left( z \right)dA\left( z \right),$$ where $d\,A\left( z \right)$ is the Lebesgue area measure on the unit disc.

Détail

DOI : 10.4153/CMB-2017-059-2

Mots-clés : 30J10, 30H10, 30H20, Bergman space, Blaschke product, Hardy space, integral mean