On compactness of commutator of multiplication and pseudodifferential operator

Abstract

In this short note we show results on the compactness of the commutator of pseudodifferential operator and operator of multiplication in both \(\mathrm{L}^2\) and \(\mathrm{L}^p\) setting. Our results use the boundedness results of pseudodifferential operators by Hwang and Hwang-Lee, and the Krasnoselskij type interpolation lemma. We use the obtained results to construct a variant of microlocal defect functionals via pseudodifferential operators and derive its localisation principle.