Continuity of weighted operators, Muckenhoupt $A_p$ weights, and Steklov problem for orthogonal polynomials

We consider weighted operators acting on $L^p(\mathbb{R}^d)$ and show that they depend continuously on the weight $w\in A_p(\mathbb{R}^d)$ in the operator topology. Then, we use this result to estimate $L^p_w(\mathbb{T})$ norm of polynomials orthogonal on the unit circle when the weight $w$ belongs to Muckenhoupt class $A_2(\mathbb{T})$ and $p>2$. The asymptotics of the polynomial entropy is obtained as an application.