On Lorentz mixed normed modulation spaces

Abstract

This paper is a study on a new kind modulation spaces $${M( P, Q) (\mathbb{R}^{d})}$$ and $${A ( P, Q, r) ( \mathbb{R}^{d})}$$ for indices in the range 1 < P < ∞, 1 ≤ Q < ∞ and 1 ≤ r < ∞, modelled on Lorentz mixed norm spaces instead of mixed norm L P spaces as the spaces $${M_{m}^{p, q}(\mathbb{R}^{d})}$$ (Feichtinger in Modulation spaces on locally compact Abelian groups, 1983; Gröchenig in Foundations of Time-Frequency Analysis. Birkh äuser, Boston, 2001), and Lorentz spaces as the spaces $${M( P, Q) (\mathbb{R}^{d})}$$ (Gürkanlıin J Math Kyoto Univ 46:595–616, 2006). First, we prove the main properties of these spaces. Later, we describe the dual spaces and determine the multiplier spaces for both of them. Moreover, we investigate the boundedness of Weyl operators and localization operators on $${M( P, Q) ( \mathbb{R}^{d})}$$ . Finally, we give an interpolation theorem for $${M( P, Q)(\mathbb{R}^{d})}$$ .