On Compactness of Commutators of Multiplications and Fourier Multipliers

We generalise results on compactness of commutators of multiplications and Fourier multiplier operators by Cordes (J Funct Anal 18:115–131, 1975) with respect to the smoothness of multiplication function. Our prime motivation has been a particular case known as the first commutation lemma—the basic tool for defining H-measures and H-distributions. We review and improve the known results in the \(\mathrm{L}^p\) setting, with \(1<p<\infty \), illustrating these results with an application, obtaining a generalised compactness by compensation result.