Continuity Properties of Multilinear Localization Operators on Modulation Spaces

Abstract

We introduce multilinear localization operators in terms of the short-time Fourier transform and multilinear Weyl pseudodifferential operators. We prove that such localization operators are in fact Weyl pseudodifferential operators whose symbols are given by the convolution between the symbol of the localization operator and the multilinear Wigner transform. To obtain such interpretation, we use the kernel theorem for the Gelfand–Shilov space \( {\mathscr {S}}^{( 1)} (\mathbb {R}^d) \) and its dual space of tempered ultra-distributions \( {\mathscr {S}}^{( 1)'} (\mathbb {R}^{2d})\). Furthermore, we study the continuity properties of the multilinear localization operators on modulation spaces. Our results extend some known results when restricted to the linear case.