Boundedness of the Hardy and the Hardy-Littlewood operators in the spaces ReH1 and BMO

The boundedness of the Hardy operator H and the Hardy-Littlewood operator IB are established, respectively, in Re H-1 and the space BMO of functions of bounded mean oscillation on the real axis R. Here the space 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 B is defined in R by equality (0.2), and the Hardy operator H is defined in R+ by equality (0.1) and its value Hf is continued to R- as an even (odd) function if the function f is even (odd). For an arbitrary function f one sets H(f) = H(f(+))+ H(f(-)), where f(+) is the even and f(-) is the odd component of f.