Remarks on Nehari's problem, matrix $A_2$ condition, and weighted bounded mean oscillation

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 $H^{\infty}$ by means of so-called {\em regular generators} -- bounded holomorphic functions $\phi$. The definition of {\em regularity} is given below, but let us mention now that 1) the following assumption on modulus of $\phi$ is sufficient for {\em regularity}: $\frac{1}{1-|\phi|^2}\in L^1(\mathbb{T})$; 2) there is no necessary and sufficient condition of {\em regularity} on bounded holomorphic $\phi$ in terms of $|\phi|$ on $\mathbb{T}$, \cite{Kh1}. This makes reasonable the attempt to find a weaker sufficient condition on $|\phi|$ 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 $\phi$. They involve the matrix $A_2$ condition from \cite{TV}.