Quasi-squares of pseudocontinuable functions

Abstract

For an inner function $\theta$ on the unit disk, let $K_{\theta }^p:=H^p\cap \theta \overline{H_0^p}$ 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 $f\in K_{\theta }^p$ to lie in $K_{\theta }^{p/2}$. (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 $K_{\theta }^p$ spaces.

16 pages