Carleson Measures, Vanishing Mean Oscillation and Critical Points

2025/09/30 by Bellavita, Carlo, Nicolau, Artur, Stylogiannis, Georgios
#30H20 #30H50 #47G10 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Primary: 30H35 #Secondary: 30H10

paper · doi:10.48550/arxiv.2509.26338

Abstract

Given a finite positive Borel measure μ in the open unit disc of the complex plane, we construct a bounded outer function E whose boundary values have vanishing mean oscillation such that |E| μ is a vanishing Carleson measure. As an application it is shown that given any function in a Hardy space, there exists a bounded analytic function in the unit disc whose boundary values have vanishing mean oscillation, with the same critical points and multiplicities.

Citations

• Analytic mappings of the unit disk with bounded compression
• Inner Functions, Möbius Distortion and Angular Derivatives
• Analytic mappings of the unit disk which almost preserve hyperbolic area
• Critical structures of inner functions
• Closed Range Integral Operators on Hardy, BMOA and Besov Spaces

Related

• Reverse Carleson measures for spaces of analytic functions
• Carleson measures for Hilbert spaces of analytic functions on the complex half-plane
• A remark on Carleson measures of domains in ℂⁿ
• Tent Carleson measures for Hardy spaces
• Carleson Measures for the Drury-Arveson Hardy space and other Besov-Sobolev spaces on Complex Balls