Operator In L2 Research Articles

𝐿_{𝑝} Fourier transforms on locally compact unimodular groups

acter group G of G given by F(X) =f Gx(a)f(a)da; when f is in L1(G)nL2(G), T: f->F is an isometric map into L2(G) which can be extended to an isometry of L2(G) with L2(G). Moreover the operation Lf of convolution byf in L2(G) is unitarily equivalent via T to multiplication, Mp by F in L2(G). In fact MF= TLfT-1. If G is abelian, any closed densely defined operator in L2(G) which commutes with the group translations is equivalent via T to a multiplication in L2(G) by a measurable function on G, (Segal [4]). In particular if f is measurable on the abelian group G and Lf is closed and densely defined with TLfT-1 = M it is natural to call F the Fourier transform of f. Since F and MP are essentially equivalent it makes sense to define the Fourier