Sobolev type theorems for an operator with singularity

Spaces of Sobolev type are discussed, which are defined by the operator with singularity: ${\cal D} = d/dx - (c/x)R$, 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 Schrödinger equation.