A GENERALIZED FOURIER TRANSFORM

Let be the operator in L²(ℝ^N):

We construct a generalized Fourier transform B_c1…cj…cN , which converts the operator into the multiplication operator i y_j, i.e.,

When c₁ = c₂ = ⋯ = c_N = 0, the operator becomes ∂/∂x_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 Schrödinger equation with a strongly singular potential.

Moreover, we show that there is the Friedrichs extension of -Δ + k/(|x|²), x ∈ 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.