Necessary Conditions for Single-Critical-Point Higher-Order Szeg\H{o} Sum Rules in OPUC

Abstract

We prove the necessity part of the higher-order Szeg\H{o} theorem on the unit circle for the single-critical-point weights $H_m(e^{i\theta})=(1-\cos\theta)^m$, $m\ge1$. If $\{\alpha_n\}_{n\ge0}$ are the Verblunsky coefficients of a nontrivial probability measure $d\mu=w(\theta)d\theta/(2\pi)+d\mu_{\mathrm s}$, then the weighted Szeg\H{o} condition $\int_0^{2\pi} (1-\cos\theta)^m\log w(\theta)\frac{d\theta}{2\pi}>-\infty$ implies $\Delta^m\alpha\in\ell^2, \,\, \alpha\in\ell^{2m+2}.$ The proof uses a finite-volume version of Yan's higher-order sum rule. The quadratic part yields the $m$-th difference energy, and the logarithmic tail yields the $\ell^{2m+2}$-control. The non-sign-definite critical terms are treated in two steps. First, the quartic principal critical block is isolated using the Yan quotient-algebra normal representative and shown to have a positive semidefinite Gram representation. Second, the remaining non-principal critical terms are controlled by the diagonal-vanishing property $\mathcal Y_{k,\mathrm{crit}}^{(m)} \in \mathfrak I_k^{\,m+1-k}, \,\, 2\le k\le m,$ together with the Breuer--Simon--Zeitouni normal form, discrete interpolation, and Young's inequality. These estimates yield a uniform finite-volume coercive bound, from which the necessity theorem follows for all $m\ge1$.