Badly Approximable Unimodular Functions in Weighted L p Spaces

Abstract

A function u on the unit circle T is said to be badly approximable in the weighted space L p(T,w)} if ¦¦u + f¦¦ L p(T,w) ≥¦L p(T,w) for all f ∊ H∞. We prove that if an unimodular function u is badly approximable in L p(T, w}) for all p ∊ (0, +∞) and some non-zero weight w, then \(\overline{u}\) is an inner function. We describe the inner functions Θ and the weights w on the unit circle T such that Θ is badly approximable in L p(T, w) for all p > 0. It turns out that, for given inner functions Θ, the class of all weights satisfying the above-mentioned condition depends only on the zero set of Θ. In other words, Θ is badly approximable in L p(T, w) for all p ∊ (0, +∞) if and only if \(\overline{B}\) is badly approximable in L p(T,w) for all p ∊ (0, +∞), where B is a Blaschke product with simple zeros and such that Θ−1(0) = B−1(0).