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Function Spaces And Multiplier Operators, R. Nillsen, S. Okada

Faculty of Informatics - Papers (Archive)

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 …