Boundedness of the Hardy and the Hardy–Littlewood operators in the spaces $\operatorname {Re}H^1$ and $\mathrm {BMO}$

The boundedness of the Hardy operator $\mathscr H$ and the Hardy–Littlewood operator $\mathscr B$ are established, respectively, in $\operatorname {Re}H^1$ and the space $\text {\textrm {BMO}}$ of functions of bounded mean oscillation on the real axis $\mathbb R$. Here the space $\operatorname {Re}H^1$ is isomorphic to the Hardy space of single-valued analytic functions $F(z)$ in the upper half-plane satisfying condition (0.3), the Hardy–Littlewood operator $\mathscr B$ is defined in $\mathbb R$ by equality (0.2), and the Hardy operator $\mathscr H$ is defined in $\mathbb R_+$ by equality (0.1) and its value $\mathscr Hf$ is continued to $\mathbb R_-$ as an even (odd) function if the function $f$ is even (odd). For an arbitrary function $f$ one sets $\mathscr H(f)=\mathscr H(f_+)+\mathscr H(f_-)$, where $f_+$ is the even and $f_-$ is the odd component of $f$.