Inversion formula and range conditions for a vector multi-interval finite Hilbert transform in $L^2$

Abstract

Given $n$ disjoint intervals $I_j$, on $\mathbb R$ together with $n$ functions $ψ_j\in L^2(I_j)$, $j=1,\dots n$, and an $n\times n$ matrix $Θ$, the problem is to find an $L^2$ solution $\vec φ= {\rm Col} (φ_1,\dots, φ_n)$, $φ_j \in L^2(I_j)$ to the linear system $χΘ\mathcal H \vec φ= \vecψ$, where $\mathcal H = {\rm diag} (\mathcal H_1 ,\dots, \mathcal H_n)$ is a matrix of finite Hilbert transforms and $χ=\text{diag}(χ_1,\dots,χ_n)$ is a matrix of the corresponding characteristic functions on $I_j$, and $\vec ψ={\rm Col} (ψ_1,\dots,ψ_n)$. Since we can interpret $χΘ\mathcal H \vec φ$ as a generalized vector multi-interval finite Hilbert transform, we call the formula for the solution as "the inversion formula" and the necessary and sufficient conditions for the existence of a solution as the "range conditions". In this paper we derive the explicit inversion formula and the range conditions in two specific cases: a) the matrix $Θ$ is symmetric and positive definite, and; b) all the entries of $Θ$ are equal to one. We also prove the uniqueness of solution, that is, that our transform is injective. When the matrix $Θ$ is positive definite, the inversion formula is given in terms of the solution of the associated matrix Riemann-Hilbert Problem. We also discuss other cases of the matrix $Θ$.