Norm inequalities for certain classes of functions and their Fourier transforms

We apply tools of interpolation theory and a commutative property of the Hilbert transform to prove necessary and sufficient conditions related to trigonometric series. These results extend and improve related theorems proven by several authors, summarized by Boas. In addition, we explore inequalities and operators, both connected to Hardy's inequalities, on certain classes of functions, including quasimonotone functions.