On Higher-Order Szegő Theorems with a Single Critical Point of Arbitrary Order

We prove the following higher-order Szegő theorem: If a measure on the unit circle has absolutely continuous part w ( θ ) and Verblunsky coefficients α with square-summable variation, then for any positive integer m , is finite if and only if α ∈ ℓ 2 m + 2 . This is the first known equivalence result of this kind in the regime of very slow decay, i.e., with ℓ p conditions with arbitrarily large p . The usual difficulty of controlling higher-order sum rules is avoided by a new test sequence approach.