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

Alexander Volberg, Peter Yuditskii|ArXiv.org|Mar 15, 2008

Holomorphic and Operator Theory15 references3 citations

TL;DR

This paper addresses the Nehari problem in the non-unique solution case by introducing a new sufficient condition for regularity of the generating function $\phi$ in the $H^\infty$-parametrization of solutions. It links regularity to weighted bounded mean oscillation and the matrix $A_2$ condition, proving that if $\phi$ satisfies a certain $A_2$-type condition on $|f - \langle f\rangle_I|^2 / (1 - |f|^2)$, then the associated Hankel operator has norm less than one. The key contribution is a direct proof of the absence of singular parts in the Herglotz representation of $w_c = (1 - |\phi_c|^2)/|1 - \phi_c|^2$ when $w \in A_2$, establishing a new criterion for regularity via harmonic extensions and Toeplitz invertibility.

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 $ϕ$. 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}: $\frac{1}{1-|ϕ|^2}\in L^1(\mathbb{T})$; 2) there is no necessary and sufficient condition of {\em regularity} on bounded holomorphic $ϕ$ in terms of $|ϕ|$ on $\mathbb{T}$, \cite{Kh1}. 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 $A_2$ condition from \cite{TV}.

Motivation & Objective

• To characterize regular solutions of the Nehari problem when the solution is non-unique.
• To identify weaker sufficient conditions on $|\phi|$ on $\mathbb{T}$ than $1/(1 - |\phi|^2) \in L^1$ for regularity of the generating function $\phi$.
• To establish necessary and sufficient conditions for regularity in terms of weighted bounded mean oscillation (BMO) and the matrix $A_2$ condition.
• To provide a direct proof of the absence of singular parts in the Herglotz representation of $w_c = (1 - |\phi_c|^2)/|1 - \phi_c|^2$ for $w \in A_2$, independent of Arov-Dym theory.

Proposed method

• Parametrize the solution set of the Nehari problem using bounded holomorphic functions $\phi \in H^\infty$ with $\|\phi\|_\infty \leq 1$, $\int_\mathbb{T} \log(1 - |\phi|) \, dm > -\infty$, and $\phi(0) = 0$.
• Introduce the unitary matrix $\mathcal{S}_\phi = \begin{bmatrix} \phi & \psi \\ \psi & f_0 \end{bmatrix}$, where $\psi$ is an outer function defined via the Herglotz integral of $\log(1 - |\phi|^2)$, and $f_0 = -\bar{\phi}\psi / \bar{\psi}$.
• Define a weighted $A_2$-type condition on $f = \phi$ involving $\sup_I \frac{1}{|I|} \int_I \frac{|f - \langle f\rangle_I|^2 + (1 - |\langle f\rangle_I|^2)}{1 - |f|^2} \, dm < \infty$, which ensures $\|\Gamma\| < 1$ for the Hankel operator $\Gamma x = P_-(f x)$.
• Use the Herglotz representation of $w = (1 - |\phi|^2)/|1 - \phi|^2$ and show that if $w \in A_2$, then $w_c = (1 - |\phi_c|^2)/|1 - \phi_c|^2$ has no singular part in its Herglotz decomposition for any real $c$, via harmonic extension and Toeplitz invertibility.
• Prove that $\Gamma_{\bar{g}_c / g_c} = e^{-ic} \Gamma_{\bar{g}_0 / g_0}$ for $g_c = (1 - \phi_c)/\psi$, and use Helson-Szegő’s theorem to show $\|\Gamma_{\bar{g}_0 / g_0}\| < 1$ when $|g_0|^2 \in A_2$, implying invertibility of the Toeplitz operator.
• Establish that $|1 - \phi_c|^2 / (1 - |\phi|^2) \in A_2$ by showing $1/g_c \in H^2$ and $g_c$ is a constant multiple of an $H^2$ function with $|h|^2 \in A_2$, thus linking the $A_2$ condition to the regularity of $\phi$.

Experimental results

Research questions

• RQ1 What weaker sufficient condition on $|\phi|$ on $\mathbb{T}$ than $1/(1 - |\phi|^2) \in L^1$ ensures regularity of the Nehari problem solution generator $\phi$?
• RQ2 Can the matrix $A_2$ condition be used to characterize regularity of $\phi$ in terms of weighted bounded mean oscillation of $f = \phi$?
• RQ3 Does the absence of a singular part in the Herglotz representation of $w_c = (1 - |\phi_c|^2)/|1 - \phi_c|^2$ follow from $w \in A_2$?
• RQ4 Is the Hankel operator norm $\|\Gamma\| < 1$ guaranteed if $f = \phi$ satisfies the $A_2$-type condition $\sup_I \frac{1}{|I|} \int_I \frac{|f - \langle f\rangle_I|^2 + (1 - |\langle f\rangle_I|^2)}{1 - |f|^2} \, dm < \infty$?
• RQ5 Can the regularity of $\phi$ be characterized via the invertibility of Toeplitz operators with unimodular symbols derived from $g_c = (1 - \phi_c)/\psi$?

Key findings

• The condition $\sup_I \frac{1}{|I|} \int_I \frac{|f - \langle f\rangle_I|^2 + (1 - |\langle f\rangle_I|^2)}{1 - |f|^2} \, dm < \infty$ for $f = \phi$ implies $\|\Gamma\| < 1$ for the Hankel operator $\Gamma x = P_-(f x)$, ensuring that $\phi$ generates a regular solution to the Nehari problem.
• If $w = (1 - |\phi|^2)/|1 - \phi|^2 \in A_2$, then the harmonic extension of $w$ leads to a function $\phi \in H^\infty$ with $\|\phi\|_\infty \leq 1$, and the associated $w_c = (1 - |\phi_c|^2)/|1 - \phi_c|^2$ has no singular part in its Herglotz representation for any real $c$, implying $w_c$ is purely absolutely continuous.
• The function $g_c = (1 - \phi_c)/\psi$ satisfies $g_c \in H^2$, and the Hankel operator $\Gamma_{\bar{g}_c / g_c}$ has norm less than 1, which implies the Toeplitz operator $T_{\bar{g}_c / g_c}$ is invertible.
• The identity $\Gamma_{\bar{g}_c / g_c} = e^{-ic} \Gamma_{\bar{g}_0 / g_0}$ holds, and since $|g_0|^2 \in A_2$ implies $\|\Gamma_{\bar{g}_0 / g_0}\| < 1$ by Helson-Szegő’s theorem, the norm condition is preserved under rotation.
• The function $1/g_c = \psi / (1 - \phi_c)$ is in $H^2$, and since $1/|g_c|^2 = \operatorname{Re} \left( \frac{1 + \phi_c}{1 - \phi_c} \right)$ has positive real part, it follows that $|1 - \phi_c|^2 / (1 - |\phi|^2) \in A_2$, providing a new $A_2$-based criterion for regularity.
• The absence of a singular part in $w_c$'s Herglotz representation is equivalent to $w_c$ being uniformly in $A_2$, which is established via the invertibility of $T_{\bar{g}_c / g_c}$ and the fact that $g_c$ is a constant multiple of an $H^2$ function with $|h|^2 \in A_2$.