## Characterizations of some properties on weighted modulation and Wiener amalgam spaces.(English)Zbl 1427.42028

Summary: In this paper, we characterize some properties on weighted modulation and Wiener amalgam spaces by the corresponding properties on weighted Lebesgue spaces. As applications, we obtain sharp conditions for product inequalities, convolution inequalities, and embedding on weighted modulation and Wiener amalgam spaces. By a unified approach different from others we give a complete answer to the question of finding sharp conditions of certain relations on weighted modulation and Wiener amalgam spaces.

### MSC:

 42B35 Function spaces arising in harmonic analysis 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 47G30 Pseudodifferential operators

### Keywords:

weighted Lebesgue spaces
Full Text:

### References:

 [1] A. Bényi, K. Grochenig, C. Heil, et al., Modulation spaces and a class of bounded multilinear pseudodifferential operators, J. Operator Theory 54 (2005), no. 2, 387-399. · Zbl 1106.47041 [2] A. Bényi, K. Gröchenig, K. A. Okoudjou, and L. G. Rogers, Unimodular Fourier multiplier for modulation spaces, J. Funct. Anal. 246 (2007), 366-384. · Zbl 1120.42010 [3] A. Bényi and K. A. Okoudjou, Local well-posedness of nonlinear dispersive equations on modulation spaces, Bull. Lond. Math. Soc. 41 (2009), no. 3, 549-558. · Zbl 1173.35115 [4] J. Chen, D. Fan, and L. Sun, Asymptotic estimates for unimodular Fourier multipliers on modulation spaces, Discrete Contin. Dyn. Syst. 32 (2012), 467-485. · Zbl 1241.42010 [5] E. Cordero and K. Gröchenig, Time-frequency analysis of localization operators, J. Funct. Anal. 205 (2003), no. 1, 107-131. · Zbl 1047.47038 [6] E. Cordero, K. Gröchenig, F. Nicola, and L. Rodino, Generalized metaplectic operators and the Schrödinger equation with a potential in the Sjöstrand class, J. Math. Phys. 55 (2014), no. 8, 081506. · Zbl 1295.81050 [7] E. Cordero and F. Nicola, Some new Strichartz estimates for the Schrödinger equation, J. Differential Equations 245 (2008), no. 7, 1945-1974. · Zbl 1154.35081 [8] E. Cordero and F. Nicola, Metaplectic representation on Wiener amalgam spaces and applications to the Schrödinger equation, J. Funct. Anal. 254 (2008), no. 2, 506-534. · Zbl 1136.22006 [9] E. Cordero and F. Nicola, Sharpness of some properties of Wiener amalgam and modulation spaces, Bull. Aust. Math. Soc. 80 (2009), no. 1, 105-116. · Zbl 1185.42021 [10] E. Cordero and F. Nicola, Remarks on Fourier multipliers and applications to the wave equation, J. Math. Anal. Appl. 353 (2009), no. 2, 583-591. · Zbl 1164.35052 [11] E. Cordero, F. Nicola, and L. Rodino, On the global boundedness of Fourier integral operators, Ann. Global Anal. Geom. 38 (2010), no. 4, 373-398. · Zbl 1200.35347 [12] J. Cunanan, M. Kobayashi, and M. Sugimoto, Inclusion relations between $$L_p$$-Sobolev and Wiener amalgam spaces, J. Funct. Anal. 268 (2015), no. 1, 239-254. · Zbl 1304.42056 [13] H. G. Feichtinger, Banach convolution algebras of Wiener type, Functions, series, operators, vol. I, II, Budapest, 1980, pp. 509-524, North-Holland, Amsterdam, 1983. [14] H. G. Feichtinger, Generalized amalgams, with applications to Fourier transform, Canad. J. Math. 42 (1990), no. 3, 395-409. · Zbl 0733.46014 [15] H. G. Feichtinger, Modulation spaces on locally compact Abelian group, Technical report, University of Vienna, 1983, Proc. internat. conf. on wavelet and applications, pp. 99-140, New Delhi Allied Publishers, India, 2003. [16] H. G. Feichtinger, Modulation spaces: looking back and ahead, Sampl. Theory Signal Image Process. 5 (2006), no. 2, 109-140. · Zbl 1156.43300 [17] H. G. Feichtinger and G. Narimani, Fourier multipliers of classical modulation spaces, Appl. Comput. Harmon. Anal. 21 (2006), 349-359. · Zbl 1106.42005 [18] Y. V. Galperin and S. Samarah, Time-frequency analysis on modulation spaces \(M^{p,q}_m, 0
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