Remarks on Nehari's problem, matrix A₂ condition, and weighted bounded mean oscillation

2008/03/15 by Alexander Volberg, Volberg, A., Peter Yuditskii +1
Mathematics · #42A50 #42C05 #42C20 #47B38 #Analytic and geometric function theory #FOS: Physical sciences #Holomorphic and Operator Theory #Mathematical Physics (math-ph) #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.0803.2245

openalex publication_date 2008/03/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider Nehari's problem in the case of non-uniqueness of solution. The solution set is then parametrized by the unit ball of H^∞ by means of so-called \em regular generators -- bounded holomorphic functions ϕ. The definition of \em regularity is given below, but let us mention now that 1) the following assumption on modulus of ϕ is sufficient for \em regularity: (1)/(1-|ϕ|²)∈ L¹(\mathbbT); 2) there is no necessary and sufficient condition of \em regularity on bounded holomorphic ϕ in terms of |ϕ| on \mathbbT, \citeKh1. This makes reasonable the attempt to find a weaker sufficient condition on |ϕ| than the condition in 1). This is done here. Also we are discussing certain new necessary and sufficient conditions of \em regularity in terms of bounded mean (weighted) oscillations of ϕ. They involve the matrix A₂ condition from \citeTV.

Citations

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