Quasi-squares of pseudocontinuable functions

TL;DR

This paper introduces a novel 'quasi-squaring' method for functions in star-invariant subspaces of Hardy spaces, enabling new extrapolation results for sublinear operators acting on these spaces.

Contribution

It proposes a new quasi-squaring operation that preserves the structure of star-invariant subspaces, overcoming limitations of the standard squaring operation.

Findings

The quasi-squaring procedure does not increase the degree as standard squaring does.

Application of the method leads to an extrapolation theorem for sublinear operators.

The approach enhances understanding of pseudocontinuable functions and their algebraic properties.

Abstract

For an inner function $\theta$ on the unit disk, let $K^p_\theta:=H^p\cap\theta\overline{H^p_0}$ be the associated star-invariant subspace of the Hardy space $H^p$. While the squaring operation $f\mapsto f^2$ maps $H^p$ into $H^{p/2}$, one cannot expect the square $f^2$ of a function $f\in K^p_\theta$ to lie in $K^{p/2}_\theta$. (Suffice it to note that if $f$ is a polynomial of degree $n$, then $f^2$ has degree $2n$ rather than $n$.) However, we come up with a certain "quasi-squaring" procedure that does not have this defect. As an application, we prove an extrapolation theorem for a class of sublinear operators acting on $K^p_\theta$ spaces.