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On a Stein And Weiss Property of the Conjugate Function

Published online by Cambridge University Press:  20 November 2018

Nakhlé Asmar
Nakhlé Asmar*
Affiliation:
University of Missouri, Columbia, Columbia, Missouri
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(1.1) The conjugate function on locally compact abelian groups. Let G be a locally compact abelian group with character group Ĝ. Let μ denote a Haar measure on G such that μ(G) = 1 if G is compact. (Unless stated otherwise, all the measures referred to below are Haar measures on the underlying groups.) Suppose that Ĝ contains a measurable order P: P + P ⊆P; PU(-P)= Ĝ; and P⋂(—P) =﹛0﹜. For ƒ in ℒ2(G), the conjugate function of f (with respect to the order P) is the function whose Fourier transform satisfies the identity for almost all χ in Ĝ, where sgnP(χ)= 0, 1, or —1, according as χ =0, χ ∈ P\\﹛0﹜, or χ ∈ (—P)\﹛0﹜.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 42 , Issue 1 , 01 February 1990 , pp. 109 - 125
Copyright
Copyright © Canadian Mathematical Society 1990

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