Geometric structure for \(\text{OpS}_{1,1}^m\) (Q1579032)

The paper under review studies pseudo-differential operators of \(\text{OpS}^m_{1,1}\). Each operator in this class coresponds to its wavelet coefficients that might be used to give a nonstandard representation of the kernel-distribution and of the symbol of the operator. The main results of the paper use this representation to provide a characterisation of \(\text{OpS}^m_{1,1}\) with a discrete space and of \(\text{OpS}^0_{1,1}\) with a kernel-distribution space. Meyer's theorem of characterisation of the operators in \(\text{OpS}^0_{1,1}\) is also obtained as a consequence. The main tool used to study such opeators consists of \textit{Y. Meyer}'s wavelets [``Ondelettes et opérateurs'', I (1990; Zbl 0694.41037), II (with \textit{R. R. Coifman}) (1991; Zbl 0745.42012)] that comes from the Beylkin-Coifman-Rokhlin algorithm [cf. \textit{G. Beylkin, R. Coifman} and \textit{V. Rokhlin}, Commun. Pure Appl. Math. 44, No. 2, 141-183 (1991; Zbl 0722.65022)].