Orthogonal Polynomials on the Circle for the Weight w Satisfying Conditions w, w-1 ε BMO

For the weight w satisfying , we prove the asymptotics of in , where are monic polynomials orthogonal with respect to w on the unit circle . 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.