Degree Name

Doctor of Philosophy


Department of Mathematics


be a locally compact group. A difference subspace of LP(G)(1 ≤ p ≤ oo) is a vector subspace of LP(G) generated by a set of vectors of the form ƒ-μ*ƒ, where ƒ is in LP(G), μ is in some prescribed set of measures on G, and * is convolution in the usual sense.

The main aim of this thesis is to present some characterisations of difference subspaces on locally compact Abelian groups. The thesis contains a generalisation of the result, obtained by B. E. Johnson, which characterised a certain type of difference subspace of L2(G) for a weakly polythetic compact Abelian group G. The thesis also extends a result of J. Bourgain on the circle group to more complicated compact Abelian groups. Moreover, there is a characterisation of certain types of difference subspaces of L2(G) for a weakly polythetic locally compact Abelian group G. This is an extension of R. Nillsen's known results on the non-compact groups Rn and Zn , where R is the group of real numbers and Z is the group of integers.

Besides the characterisation of difference subspaces, the thesis contains some applications of difference subspaces. There is some discussion on the relationships between the behaviour of functions in difference subspaces on a compact Abelian group and the minimum number of elements required to generate the group topologically. Furthermore, there is an investigation of the relationships between the difference subspaces and the ranges of linear differential operators on connected Abelian Lie groups.