Abstract
For an inner function θ on the unit disk, let Kθp:=Hp∩θH0p‾ be the associated star-invariant subspace of the Hardy space Hp. While the squaring operation f↦f2 maps Hp into Hp/2, one cannot expect the square f2 of a function f∈Kθp to lie in Kθp/2. (Suffice it to note that if f is a polynomial of degree n, then f2 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 Kθp spaces.
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