Blaschke products and nonideal ideals in higher order Lipschitz algebras

TL;DR

This paper explores the structure of certain ideals in higher order Lipschitz algebras related to Blaschke products, revealing they are not characterized solely by size conditions on functions.

Contribution

It introduces new results on the non-ideal nature of specific Blaschke product-associated ideals in higher order Lipschitz algebras and provides a novel factorization theorem for $H^ abla_n$.

Findings

Ideals associated with Blaschke products in $A^ abla$ are not ideal spaces.

New canonical factorization theorem for $H^ abla_n$.

Analysis of these ideals for integer $ abla=n$ in $H^ abla_n$.

Abstract

We investigate certain ideals (associated with Blaschke products) of the analytic Lipschitz algebra $A^\alpha$, with $\alpha>1$, that fail to be "ideal spaces". The latter means that the ideals in question are not describable by any size condition on the function's modulus. In the case where $\alpha=n$ is an integer, we study this phenomenon for the algebra $H^\infty_n=\{f:f^{(n)}\in H^\infty\}$ rather than for its more manageable Zygmund-type version. This part is based on a new theorem concerning the canonical factorization in $H^\infty_n$.