Weak embedding property, inner functions and entropy

• Published in: Mathematische annalen Vol. 368; no. 3-4; pp. 987 - 1015
• Main Authors: Borichev, Alexander, Nicolau, Artur, Thomas, Pascal J.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2017; Springer Nature B.V
• Subjects: Constrictions; Decay rate; Entropy; Equivalence; Mathematics; Mathematics and Statistics; Multiplication; Porosity; 30J15; 30H05; 30H80; 30J05
• ISSN: 0025-5831; 1432-1807
• DOI: 10.1007/s00208-016-1464-4

Following Gorkin, Mortini, and Nikolski, we say that an inner function I in H ∞ ( 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 ∞ / I H ∞ . 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.