Vol. 26, No. 2, 1968

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ISSN: 0030-8730
On measures with small transforms

Raouf Doss

Vol. 26 (1968), No. 2, 257–263
Abstract

G is a locally compact abelian group whose dual Γ is algebraically ordered, i.e., ordered when considered as a discrete group. Every (Radon) complex measure μ on G has a unique Lebesgue decomposition: = s + g(x)dx, where s is singular and g L1(G). A measure μ on G is of analytic type if μ(γ) = 0 for γ < 0, where μ is the Fourier-Stieltjes transform of μ.

The main result of the paper is that if γ<0|μ(γ)|2 dγ < , or more generally, if, for γ < 0, μ(γ) coincides with the transform f(γ) of a function f in Lp(G), 1 p 2, then the singular part s is of analytic type and μs(0) = 0.

Mathematical Subject Classification
Primary: 42.52
Milestones
Received: 27 November 1967
Revised: 27 March 1968
Published: 1 August 1968
Authors
Raouf Doss