The Steklov problem and estimates for orthogonal polynomials with $A_p(\mathbb{T})$ weights

The presented paper is devoted to bounds of the orthogonal polynomials on the support of the measure of orthogonality. The big interest to this subject-matter is caused by famous Steklov problem and it's modern development and understanding. We consider weights on the unit circle $\mathbb{T}$ with $A_p$ characteristic close to $1$. For the corresponding orthonormal polynomials, we obtain the upper estimates on the weighted $L^p$ norm with $p\in (2,\infty]$.