s-numbers of singular integrals for the invariance of absolutely continuous spectra in fractional dimensions
Let K be a \(C^{\infty}\)-function on \({\mathbb{R}}^ n\setminus \{0\}\) such that \(K(\lambda x)=\lambda^{-\alpha}K(x)\) for some \(\alpha >0\). Let \(\mu\) be a compactly supported positive Radon measure on \({\mathbb{R}}^ n\), and define \[ h(r):=\sup \{\mu (B(x,r)):\;x\in {\mathbb{R}}^ n\}, \] where B(x,r) is the open ball with center x and radius \(r>0\). Then \[ T: f(x)\to \int K(x-y)f(y)d\mu (y) \] defines a compact operator on \(L^ 2({\mathbb{R}}^ n,\mu)\). The authors show how the asymptotic behaviour of the s-numbers \(s_ n(T)\) depends on \(\alpha\) and on the behaviour of h(r) as \(r\to 0\). An application to the kernels \(K_ j(x)=x_ j/| x|^ 2\) yields results about the invariance of the absolutely continuous spectrum of the n-tuple of multiplication operators \(f(x)\to x_ jf(x)\) with respect to perturbation by operators belonging to certain normed ideals. The paper is readable for insiders only.
compactly supported positive Radon measure
scientific article; zbMATH DE number 4210878
scientific article