Hardy–Goldberg operator and its conjugate one in Hardy spaces and $BMO(\mathbb T)$

The Hardy operator transforming a sequence of Fourier coefficients of a function to a sequence of its arithmetic means is well-known in harmonic analysis. In the present paper we consider the Hardy–Goldberg operator generalizing Hardy operator and its conjugate operator. We prove the boundedness of Hardy–Goldberg operator in real Hardy space and of its analog in Hardy space on disc. We establish the boundedness of conjugate Hardy–Goldberg operator in periodic $BMO$ and $VMO$ operators.