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 \({M( P, Q)(\mathbb{R}^{d})}\).
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Acknowledgments
The author would like to thank Elena Cordero, Luigi Rodino, Joachim Toft, M. W. Wong and Patrik Wahlberg for fruitful discussions and comments. This work was started while she was visiting the Department of Mathematics of the University of Torino. She also wants to thank Elena Cordero and Luigi Rodino for their wonderful hospitality and for great working conditions.
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Sandıkçı, A. On Lorentz mixed normed modulation spaces. J. Pseudo-Differ. Oper. Appl. 3, 263–281 (2012). https://doi.org/10.1007/s11868-012-0051-z
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DOI: https://doi.org/10.1007/s11868-012-0051-z