Function spaces and multiplier operators

Let G denote a locally compact Hausdorff abelian group. Then a bounded linear operator T from L^2(G) into L^2(G) is a bounded multiplier operator if, under the Fourier transform on L^2(G ), for each function f in L^2(G), T(f) changes into a bounded function U times the Fourier transform of f. Then U is called the multiplier of T. An unbounded multiplier operator has a similar definition, but its domain is a dense subspace of L^2(G) and the multiplier function need not be bounded. For example, differentiation on the first order Sobolev subspace of L^2(R) is an unbounded multiplier operator with multiplier mapping x into ix, and the Laplace operator on the second order Sobolev subspace of L^2(R^2) is an unbounded multiplier operator with multiplier mapping (x,y) into –x^2-y^2. Now in 1972 ( J. Functional Analysis 11, pp.407-424), G. Meisters and W. Schmidt had effectively characterized the range of the differentiation operator on the first order Sobolev subspace of L^2(-pi,pi) as a space of first order differences. Corresponding results for the real line, the Laplace operator in n dimensions, and other differential operators were obtained by the first named author (see Springer Lecture Notes in Mathematics, vol. 1586, for example). The present paper extends earlier results so that they apply to general bounded or unbounded multiplier operators on the space L^2(G) and on certain spaces of abstract distributions on G. Descriptions of the ranges of such operators are obtained, and these ranges are either Banach or Hilbert spaces in weighted L^p or L^2 norms under the Fourier transform, and the operators become isometries from their domains onto these spaces. These spaces, that is the ranges of the operators, are described in terms of finite differences involving pseudomeasures on the group.