If K is a local field, the
Hardy space H1(K) is defined as follows: If f is a distribution on K let f(x,k)
(defined on K ×Z) be its regularization. Let f∗(x) =supk|f(x,k)|. Then f ∈ H1 iff
the maximal function f∗ is integrable. Chao has given the following conjugate
function characterization of H1. Let π be a multiplicative character on K
that is homogeneous of degree zero, ramified of degree 1, and is odd. Then
f ∈ L1 is in H1 iff (πf)∨∈ L1. He also shows that if μ is a finite (Borel)
measure then μ is absolutely continuous whenever (μπ)∨ is also a finite
measure. In this paper proofs are given that these results fail if π is not
odd.
