On Lorentz mixed normed modulation spaces (Q1941964)

The author introduces a new type of modulation spaces, called \(M(P, Q)(\mathbb R^d)\) and \(A(P, Q, r)(\mathbb R^d)\), modelled on Lorentz mixed normed spaces rather than mixed normed spaces \(L^P (\mathbb R^d)\) as the \textit{Feichtinger's} ones [``Modulation spaces on locally compact abelian groups'', Technical Report, University of Vienna, (1983)] and the Lorentz spaces as the \textit{Gürkanli's} ones [J. Math. Kyoto Univ. 46, No. 3, 595--616 (2006; Zbl 1155.46012)]. In section 2, the author shows that the space \(M(P, Q)(\mathbb R^d)\) is a Banach space, and that the definition is independent of the choice of the window functions. She also studies the inclusion relations and the dual of this space. Moreover by using estimates on the cross-Wigner distribution, she works on the boundedness of the Weyl operators and the localization operators on the Lorentz mixed normed modulation space \(M(P, Q)(\mathbb R^d)\). In section 3, the author defines the space \(A(P, Q, r)(\mathbb R^d)\) and discusses some similar properties as in section 2. Also she shows that if \(r = 1\), the space \(A(P, Q, r)(\mathbb R^d)\) is a Segal algebra. Finally, in section 4, the author focuses on multipliers spaces of \(M(P, Q)(\mathbb R^d)\) and \(A(P, Q, r)(\mathbb R^d)\).