An embedding theorem of Sobolev type for an operator with singularity

Watanabe, Shuji

doi:10.1090/S0002-9939-97-03642-3

An embedding theorem of Sobolev type for an operator with singularity
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Proc. Amer. Math. Soc. 125 (1997), 839-848 Request permission

Abstract:

We discuss spaces of Sobolev type 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. We study smoothness of $u$ and continuity of $u / x^{\beta }$ ($\beta > 0$) where $u$ is in each space of Sobolev type, and obtain a generalization of the Sobolev embedding theorem. On the basis of a generalization of the Fourier transform, the proof is carried out. We apply the result to the Cauchy problems for partial differential equations with singular coefficients.

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