Hardy and Bellman Transformations in Spaces Close to L∞ and L1

Abstract

The replacement of coefficients of a trigonometric series by their arithmetic averages gives rise to the Hardy operator. The Bellman operator is its adjoint. The spaces Lp with p∈[1,∞) are invariant under the Hardy transformation. This result was proved by Hardy. On the other hand, the space L∞ is not invariant under the Hardy transformation, and the space L1 is not invariant under the Bellman transformation. Golubov proved that the space BMO is not invariant under the Hardy transformation and \({\text{Re}}\;^{\text{ + }} \;H\) 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, and Marcinkiewicz spaces and the spaces BMO and \({\text{Re}}\;^{\text{ + }} \;H\). For the Hardy operator, this shift occurs if the domain is close to L∞, and for the Bellman operator the same happens if the domain is close to L1. Bibliography: 15 titles.