Isomorphic copies of l_∞  in the weighted Hardy spaces on the unit disc.

In this lecture, we shall discuss a question whether the density of analytic polynomials in an H-admissible space is sufficient to the minimality of the space? This question has a purely foundational background, relating fundamental concepts from the theory of H^p spaces. We solve this problem by finding suitable counterexamples of Hardy spaces built upon some weighted Lebesgue spaces. In particular, we provide a direct construction of weights from Szegö’s class, which guarantees the existence of isomorphic copies of l_∞ in weighted Hardy spaces on the unit disc.