The Stein-Weiss theorem that the distribution function of the Hilbert transform of the characteristic function of E depends only on the measure of E is generalized for the ergodic Hilbert transform in the case of a one-parameter flow of measure-preserving transformations on a σ-finite measure space.
The Stein-Weiss theorem that the distribution function of the Hilbert transform of the characteristic function of E depends only on the measure of E is generalized to the ergodic Hilbert transform.
The uniqueness theorem for the ergodic maximal operator is proved in the continous case.
It is proved that the ergodic maximal operator is one-to-one.
A weighted ergodic maximal equality is proved for a conservative and ergodic semiflow of nonsingular automorphisms.
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