Let Γ be a dense subgroup of
the real line ℝ. Endow Γ with the discrete topology and the order it inherits from ℝ,
and let K be the dual group of Γ. Helson’s classic theory of generalized analyticity
uses the spectral decomposability of unitary groups to establish a one-to-one
correspondence between the cocycles on K and the normalized simply invariant
subspaces of L2(K). This theory has been extended to the invariant subspaces of
Lp(K),1 < p < ∞, by using recent results concerning the spectral decomposability
of uniformly bounded one-parameter groups acting on UMD spaces. We show here
that each cocycle A on K can be used to transfer the classical Hilbert transform from
L1(ℝ) to L1(K) in terms of almost everywhere convergence on K so that in the
interesting case (i.e., when A is not a coboundary) the corresponding invariant
subspace of Lp(K) is a generalized ergodic Hardy space. This description of the
invariant subspaces explicitly identifies the role of the Hilbert transform in
generalized analyticity on K. The formulation in terms of almost everywhere
convergence on K provides an intrinsic viewpoint which extends to the case
p = 1.
