Sobolev type theorems for an operator with singularity

Spaces of Sobolev type are discussed, which are defined by the operator with singularity: D = d/dx - (c/x)R, where Ru(x) = u(-x) and c > 1. This operator appears in a one-dimensional harmonic oscillator governed by Wigner's commutation relations. Smoothness of u and continuity of u/x(beta) (beta > 0) are studied where u is in each space of Sobolev type, and results similar to Sobolev's lemma are obtained. The proofs are carried out based on a generalization of the Fourier transform. The results are applied to the Schrodinger equation.