Hardy and Bellman transformations in spaces close to $L_\infty$ and to $L_1$

Replacement of the coefficients of a trigonometric series by their arithmetic averages gives rise to the Hardy operator. The Bellman operator is its adjoint. The spaces $L_p$ with $p\in[1,\infty)$ are invariant under the Hardy transformation. This result was proved by Hardy. On the other hand, the space $L_\infty$ is not invariant under the Hardy transformation and $L_1$ is not invariant under the Bellman transformation. B. I. Golubov has proved that the space BMO is not invariant under the Hardy transformation and $\operatorname{Re}^{+}H$ is not invariant under the Bellman operator. In the present paper the exact “shift” of the domain under the action of these operators is described for certain Orlicz, Lorenz, Marcinkiewicz spaces, BMO, and $\operatorname{Re}^{+}H$. For the Hardy operator this shift occurs if the domain is close to $L_\infty$, and for the Bellman operator if the domain is close to $L_1$.