Properties of extremal elements in the duality relation for Hardy spaces

Consider a Hardy space Hp in the unit disk D, p≥1. Let lω be a linear functional on Hp determined by ω∈Lq (T=∂D, 1/p+1/q=1) and let F be an extremal function for lω. Let X∈Hq implements the best approximation of ω¯ in Lq(T) by functions from H0q={y∈Hq:y(0)=0}. The functions F and X are called extremal elements (e. e.) for lω. E. e. are related by the corresponding duality relation.We consider the problem of how certain properties of ω will affect e. e. An article by L. Carleson and S. Jacobs (1972), investigated the problem of the properties of elements on which the infimum inf{∥ω¯-x∥L∞(T): x∈H0∞} for a given ω∈Lq(T) is attained. The hypothesis of the authors that the relationship between extremal elements is similar to that of the function ω and its projection onto Hq is partially confirmed in a paper by V. G. Ryabykh (2006). Some properties of e. e. for lω, when ω is a polynomial, were studied in a paper by Kh. Kh Burchaev, G. Yu. Ryabykh V. G. Ryabykh (2017). In this paper, relying on the main result of the last article and using the method of successive approximations, the following is proved: if ω∈Lq∗(T) and q≤q∗