On Hankel-type operators with discontinuous symbols in higher dimensions
For a symbol \(a\) and \(\alpha\geq 1\), \(d\geq 2\), let \(\mathrm{Op}_\alpha(a)\) be the pseudodifferential operator defined by NEWLINE\[NEWLINE (\mathrm{Op}_\alpha a)u(x)=\left(\frac{\alpha}{2\pi}\right)^d\int_{\mathbb{R}^d}\int_{\mathbb{R}^d} e^{i\alpha(x-y)\xi}a(x,\xi)u(y)\,d\xi\,dy. NEWLINE\]NEWLINE For bounded domains \(\Lambda\) and \(\Omega\) in \(\mathbb{R}^d\), consider the Hankel-type pseudodifferential operators with discontinuous symbols defined by NEWLINE\[NEWLINE G_\alpha(a)=(1-\chi_\Lambda) \mathrm{Op}_\alpha(\chi_\Omega) \mathrm{Op}_\alpha(a) \mathrm{Op}_\alpha(\chi_\Omega)\chi_\Lambda, \quad H_\alpha(a)=G_\alpha(a)+G_\alpha^*(a). NEWLINE\]NEWLINE For a symbol \(a\) and any \(C^1\)-surfaces \(S\) and \(P\), put NEWLINE\[NEWLINE \mathcal{M}_1(a)=\frac{1}{(2\pi)^{d-1}}\int_S\int_P a(x,\xi)|\mathbf{n}_S(x)\cdot\mathbf{n}_P(\xi)|\,dS_\xi\,dS_x, NEWLINE\]NEWLINE where \(\mathbf{n}_S(x)\) and \(\mathbf{n}_P(\xi)\) denote the exterior normals to \(S\) and \(P\) at the points \(x\) and \(\xi\), respectively. For \(b\in(0,\infty)\) and any function \(g\) on \(\mathbb{R}\) such that \(g(t)=O(|t|)\), define the integral NEWLINE\[NEWLINE \mathcal{U}(g,b)=\frac{2}{\pi^2}\int_0^1\frac{g(bt/2)}{t\sqrt{1-t^2}}\, dt NEWLINE\]NEWLINE and denote \(g_{ev}(t)=(g(t)+g(-t))/2\).NEWLINENEWLINEThe main result of the paper is the following. Let \(\Lambda,\Omega\subset\mathbb{R}^d\) be bounded domains in \(\mathbb{R}^d\) such that \(\Lambda\) is \(C^1\) and \(\Omega\) is \(C^3\). Let \(a=a(x,\xi)\) be a symbol satisfying the condition NEWLINE\[NEWLINE\max_{0\leq n,m\leq d+2}\sup_{x,\xi}|\nabla_x^n\nabla_\xi^m a(x,\xi)|<\inftyNEWLINE\]NEWLINE and having a compact support in both variables. Let \(g\) be a function on \(\mathbb{R}\) such that \(g_{ev}(t)t^{-2}\) is continuous on \(\mathbb{R}\). Then NEWLINE\[NEWLINE \text{trace}\, g(H_\alpha(a))= \alpha^{d-1}\log\alpha\, \mathcal{M}_1(\mathcal{U}(g_{ev};|a|);\partial\Lambda,\partial\Omega) +o(\alpha^{d-1}\log\alpha) NEWLINE\]NEWLINE as \(\alpha\to\infty\).
scientific article; zbMATH DE number 6048035