Let G be a locally compact abelian group, with character group Ĝ. Let ψ be an arbitrary continuous real-valued homomorphism defined on Ĝ. For f in LP(G), 1 < p ≤ 2, let
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0004972700017494/resource/name/S0004972700017494_eqnU1.gif?pub-status=live)
where 1[−ν, ν] is the indicator function of the interval [ − ν, ν ], and I is an unbounded increasing sequence of positive real numbers. Then there is a constant Mp, independent of f, such that ‖M#f‖p ≤ Mp ‖f‖p. Consequently, the pointwise limit of the function
exists, almost everywhere on G, as ν tends to infinity. Using this result and a generalised version of Riesz's theorem on conjugate functions, we obtain a pointwise inversion for Fourier transforms of functions on Ra × Tb, where a and b are nonnegative integers, and on various other locally compact abelian groups.