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Exponential Estimates for the Conjugate Function on Locally Compact Abelian Groups

Published online by Cambridge University Press:  20 November 2018

Nakhlé Habib Asmar
Nakhlé Habib Asmar*
Affiliation:
Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211 U.S.A.
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Abstract

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Let G be a locally compact Abelian group, with character group X. Suppose that X contains a measurable order P. For the conjugate function of f is the function whose Fourier transform satisfies the identity for almost all χ in X where sgnp(χ) = - 1 , 0, 1, according as We prove that, when f is bounded with compact support, the conjugate function satisfies some weak type inequalities similar to those of the Hilbert transform of a bounded function with compact support in ℝ. As a consequence of these inequalities, we prove that possesses strong integrability properties, whenever f is bounded and G is compact. In particular, we show that, when G is compact and f is continuous on G, the function is integrable for all p > 0.

Keywords

43A17

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 33 , Issue 1 , 01 March 1990, pp. 34 - 44
Copyright
Copyright © Canadian Mathematical Society 1990

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