\(L^p\)-nuclear pseudo-differential operators on \(\mathbb Z\) and \(\mathbb S^1\) (Q2846921)

The authors consider pseudo-differential operators on the unit circle \(\mathbb{S}^1\) of the form NEWLINE\[NEWLINEPf(x)= \sum^{+\infty}_{n=-\infty} e^{inx} \sigma(x,n)\,\widehat f(n),\quad x\in [-\pi,\pi],NEWLINE\]NEWLINE where \(\widehat f\) is the Fourier transform of \(f\). Sufficient conditions on the symbol \(\sigma\) in order to ensure the boundedness of \(P: L^p(\mathbb{S}^1)\to L^p(\mathbb{S}^1)\), \(1\leq p<\infty\), were given by \textit{S. Molahajloo} and \textit{M. W. Wong} [Operator Theory: Advances and Applications 189, 297--306 (2009; Zbl 1210.47073)].NEWLINENEWLINE In the present paper, the authors give sufficient conditions on \(\sigma\) for the nuclearity of \(P: L^{p_1}(\mathbb{S}^1)\to L^{p_2}(\mathbb{S}^1)\), \(1\leq p_1\), \(p_2<\infty\).