Orthogonal Polynomials on the Circle for the Weight w Satisfying Conditions $$w,w^{-1}\in \mathrm{BMO}$$ w , w - 1 ∈ BMO

Abstract

For the weight w satisfying \(w,w^{-1}\in \mathrm{BMO}({\mathbb {T}})\), we prove the asymptotics of \(\{\Phi _n(e^{i\theta },w)\}\) in \(L^p[-\pi ,\pi ], 2\leqslant p<p_0\), where \(\{\Phi _n(z,w)\}\) are monic polynomials orthogonal with respect to w on the unit circle \({\mathbb {T}}\). Immediate applications include the estimates on the uniform norm and asymptotics of the polynomial entropies. The estimates on higher-order commutators between the Calderon–Zygmund operators and BMO functions play the key role in the proofs of main results.