The ergodic Hilbert transform on the weighted spaces \({\mathfrak L}_p (G,w)\) (Q1293297)

The author considers the continuity of the ergodic Hilbert transform from a weighted \(L_F\)-space \(L_F(G,w)\) into itself, where \(G\) is a locally compact abelian group. The uniform \(A_F\)-conditions are defined. The main result is as follows: The ergodic Hilbert transform is bounded uniformly with respect to continuous homomorphisms from \(R\) into \(G\) exactly if a continuous weight \(w\) satisfies the uniform \(A_F\)-conditions. -- This reduces to the classical result for the cases of \(G=R\) or \(T\).