Multiplication and division in the space of analytic functions with area integrable derivative, and in some related spaces

S. A. Vinogradov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 23, Tome 222 (1995), pp. 45-77

Résumé

We study the possibility of multiplication and division by inner functions (in the sense of Beurling) in $A^1_p$ ($0$), the space of functions analytic in the unit disk $\mathbb D$ and such that $$ \int_\mathbb D|f'(z)|^p(1-|z|)^{p-1}\,dm_2(z)+\infty $$ ($m_2$ is the planar Lebesgue measure). In particular, a simple description is given for multipliers of the space $A^1_p$ for $p\in(0,2)$. Conditions on zeros for the Blaschke products are given under which a product is a multiplier or a divisor in $A^1_p$ ($0$). It is shown that the singular function $\exp\frac{z+1}{z-1}$ is a multiplier but not a divisor in the space $A^1_p$ ($0$). Bibliography: 17 titles.