Weak embedding property, inner functions and entropy

Abstract

Following Gorkin, Mortini, and Nikolski, we say that an inner function I in $H^\infty (\mathbb {D})$ has the WEP property if its modulus at a point z is bounded from below by a function of the distance from z to the zero set of I. This is equivalent to a number of properties, and we establish some consequences of this for $H^\infty /IH^\infty$. The bulk of the paper is devoted to wepable functions, i.e. those inner functions which can be made WEP after multiplication by a suitable Blaschke product. We prove that a closed subset E of the unit circle is of finite entropy (i.e. is a Beurling–Carleson set) if and only if any singular measure supported on E gives rise to a wepable singular inner function. As a corollary, we see that singular measures which spread their mass too evenly cannot give rise to wepable singular inner functions. Furthermore, we prove that the stronger property of porosity of E is equivalent to a stronger form of wepability (easy wepability) for the singular inner functions with support in E. Finally, we find out the critical decay rate of masses of atomic measures (with no restrictions on support) guaranteeing that the corresponding singular inner functions are easily wepable.