Sobolev type theorems for an operator with singularity

Watanabe, Shuji

doi:10.1090/S0002-9939-97-03523-5

Sobolev type theorems for an operator with singularity
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Proc. Amer. Math. Soc. 125 (1997), 129-136 Request permission

Abstract:

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

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