A generalized Fourier transform

Let D-cj,D- j be the operator in L-2(R-N): D-cj,D- j = partial derivative/partial derivativex(j) - c(j)/x(j) R-j, R(j)u(x(1),..., x(j-1), x(j), x(j+1),..., x(N)) = u(x(1),..., x(j-1), -x(j), x(j+1),..., x(N)), where c(j) > -1/2 and j = 1, 2,..., N. We construct a generalized Fourier transform B-c1...cj...cN, which converts the operator D-cj,D- j into the multiplication operator i y(j), i.e., B-c1...cj...cN D-cj, j B-c1...cj...cN(*) = i y(j). Here B-c1...cj...cN(*) is the adjoint operator of B-c1...cj...cN and i = root-1. When c(1) = c(2) = ... = c(N) = 0, the operator D-cj,D-j becomes partial derivative/partial derivativex(j). Hence the transform B-c1...cj...cN coincides with the Fourier transform. We can therefore regard the transform B-c1...cj..cN as a generalized Fourier transform. On the basis of the transform we explicitly find out the solutions of the Cauchy problems for the heat equation with a strongly singular coefficient and for the Schrodinger equation with a strongly singular potential. Moreover, we show that there is the Friedrichs extension of -Delta+ k/(\x\(2)), x is an element of R-N as long as k > -N/4. Using the transform above we define spaces of Sobolev type. Each space is a generalized Sobolev space. We show an embedding theorem for these spaces. We see that the embedding theorem is a generalization of the Sobolev embedding theorem. We finally apply the embedding theorem to the Cauchy problem for the wave equation with a strongly singular coefficient and study some properties of its solution.