Factorization of a class of Toeplitz + hankel operators and the A_p-condition

Abstract

Let $M(\varphi )=T(\varphi )+H(\varphi )$ be the Toeplitz plus Hankel operator acting on $H^p(\T)$ with generating function $ϕ\in L^\iy(\T)$. In a previous paper we proved that $M(\varphi )$ is invertible if and only if $\varphi$ admits a factorization $\varphi (t)={\varphi }_{-}(t){\varphi }_0(t)$ such that ${\varphi }_{-}$ and ${\varphi }_0$ and their inverses belong to certain function spaces and such that a further condition formulated in terms of ${\varphi }_{-}$ and ${\varphi }_0$ is satisfied. In this paper we prove that this additional condition is equivalent to the Hunt-Muckenhoupt-Wheeden condition (or, $A_p$-condition) for a certain function $\sigma$ defined on $[-1,1]$, which is given in terms of ${\varphi }_0$. As an application, a necessary and sufficient criteria for the invertibility of $M(\varphi )$ with piecewise continuous functions $\varphi$ is proved directly. Fredholm criteria are obtained as well.