On continuous linear functionals in some spaces of functions analytic in disk

The issue on description of linear continuous functionals on the spaces of analytic functions has been studied since the middle of 20th century. Historically, the structure of linear continuous functionals on the Hardy spaces $H^p$ for $p\geq 1$ was first established by A. Taylor in 1951. In the spaces $H^p$, $0<p<1$, this problem was solved by P. Duren, B. Romberg, A. Shields in 1969. We note that an estimate for the coefficient multipliers in these spaces was employed in the proof. In the present paper, by developing the method proposed in the work by P. Duren et al, we describe linear continuous functionals on area Privalov classes and Nevanlinna-Dzhrbashjan type spaces. The considered classes generalize the area Nevanlinna classes well-known in scientific literature. The idea of the proof of the main result is as follows: the issue on finding the general form of a continuous linear functional is reduced to finding a form of an arbitrary coefficient multiplier acting from a studied space into the space of bounded analytic functions. The latter problem in a simplified form can be formulated as follows: by what factors we should multiply the Taylor coefficients of the functions in a studied class in order to make them Taylor coefficients of some bounded analytic function.