Nakhlé Asmar and Stephen Montgomery-Smith, Hahn's Embedding Theorem for orders and harmonic analysis on groups with ordered duals.
Colloq. Math. 70, (1996), 235-252.
Let \(G\) be a locally compact abelian group whose dual group \(\Gamma\) contains a Haar measurable order \(P\). Using the order \(P\) we define the conjugate function operator on \(L^p(G)\), \(1 \le p < \infty\), as was done by Helson. We will show how to use Hahn's Embedding Theorem for orders and the ergodic Hilbert transform to study the conjugate function. Our approach enables us to define a filtration of the Borel \(\sigma\)-algebra on \(G\), which in turn will allow us to introduce tools from martingale theory into the analysis on groups with ordered duals. We illustrate our methods by describing a concrete way to construct the conjugate function in \(L^p(G)\). This construction is in terms of an unconditionally convergent conjugate series whose individual terms are constructed from specific ergodic Hilbert transforms. We also present a study of the square function associated with the conjugate series.
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