On measures with small transforms

Raouf Doss

Vol. 26 (1968), No. 2, 257–263

DOI: 10.2140/pjm.1968.26.257

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: dμ = dμ_s + g(x)dx, where dμ_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 L^p(G), 1 ≦ p ≦ 2, then the singular part dμ_s is of analytic type and μ_s(0) = 0.

Mathematical Subject Classification

Primary: 42.52

Milestones

Received: 27 November 1967

Revised: 27 March 1968

Published: 1 August 1968