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

We prove the following higher-order Szego theorem: If a measure on the unit circle has absolutely continuous part w(theta) and Verblunsky coefficients a with square-summable variation, then for any positive integer m, integral(1 - cos theta)(m) log w(theta)d theta is finite if and only if alpha is an element of l(2m+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.