Vol. 74, No. 2, 1978

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The failure of even conjugate characterizations of H1 on local fields

Mitchell Herbert Taibleson

Vol. 74 (1978), No. 2, 501–506

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.

Mathematical Subject Classification 2000
Primary: 43A70
Secondary: 12B40
Received: 10 August 1976
Published: 1 February 1978
Mitchell Herbert Taibleson