Open Access
January 2018 Hörmander-type theorems on unimodular multipliers and applications to modulation spaces
Qiang Huang, Jiecheng Chen, Dashan Fan, Xiangrong Zhu
Banach J. Math. Anal. 12(1): 85-103 (January 2018). DOI: 10.1215/17358787-2017-0037
Abstract

In this article, for the unimodular multipliers eiμ(D), we establish two Hörmander-type multiplier theorems by assuming conditions on their phase functions μ. As applications, we obtain two multiplier theorems particularly fitting for the modulation spaces, thus allowing us to extend and improve some known results.

References

1.

[1] A. Bényi, K. Gröchenig, K. A. Okoudjou, and L. G. Rogers,Unimodular Fourier multipliers for modulation spaces, J. Funct. Anal.246(2007), no. 2, 366–384. 1120.42010 10.1016/j.jfa.2006.12.019[1] A. Bényi, K. Gröchenig, K. A. Okoudjou, and L. G. Rogers,Unimodular Fourier multipliers for modulation spaces, J. Funct. Anal.246(2007), no. 2, 366–384. 1120.42010 10.1016/j.jfa.2006.12.019

2.

[2] A. Bényi and T. Oh,Modulation spaces, Wiener amalgam spaces, and Brownian motions, Adv. Math.228(2011), no. 5, 2943–2981. 1229.42021 10.1016/j.aim.2011.07.023[2] A. Bényi and T. Oh,Modulation spaces, Wiener amalgam spaces, and Brownian motions, Adv. Math.228(2011), no. 5, 2943–2981. 1229.42021 10.1016/j.aim.2011.07.023

3.

[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. 1173.35115 10.1112/blms/bdp027[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. 1173.35115 10.1112/blms/bdp027

4.

[4] J. Bergh and J. Löfström,Interpolation Spaces: An Introduction, Grundlehren Math. Wiss.223, Springer, Berlin, 1976.[4] J. Bergh and J. Löfström,Interpolation Spaces: An Introduction, Grundlehren Math. Wiss.223, Springer, Berlin, 1976.

5.

[5] D. G. Bhimani,The Cauchy problem for the Hartree type equation in modulation spaces, Nonlinear Anal.130(2016), 190–201. 1330.35394 10.1016/j.na.2015.10.002[5] D. G. Bhimani,The Cauchy problem for the Hartree type equation in modulation spaces, Nonlinear Anal.130(2016), 190–201. 1330.35394 10.1016/j.na.2015.10.002

6.

[6] J. Chen, D. Fan, and L. Sun,Asymptotic estimates for unimodular Fourier multipliers on modulation spaces, Discret. Contin. Dyn. Syst.32(2012), no. 2, 467–485. 1241.42010[6] J. Chen, D. Fan, and L. Sun,Asymptotic estimates for unimodular Fourier multipliers on modulation spaces, Discret. Contin. Dyn. Syst.32(2012), no. 2, 467–485. 1241.42010

7.

[7] E. Cordero, F. Nicola, and L. Rodino,Wave packet analysis of Schrödinger equations in analytic function spaces, Adv. Math.278(2015), 182–209. 1318.35094 10.1016/j.aim.2015.03.014[7] E. Cordero, F. Nicola, and L. Rodino,Wave packet analysis of Schrödinger equations in analytic function spaces, Adv. Math.278(2015), 182–209. 1318.35094 10.1016/j.aim.2015.03.014

8.

[8] J. Cunanan,On $L^{p}$-boundedness of pseudo-differential operators of Sjöstrand’s class, J. Fourier Anal. Appl.23(2017), no. 4, 810–816.[8] J. Cunanan,On $L^{p}$-boundedness of pseudo-differential operators of Sjöstrand’s class, J. Fourier Anal. Appl.23(2017), no. 4, 810–816.

9.

[9] J. Cunanan and M. Sugimoto,Unimodular Fourier multipliers on Wiener amalgam spaces, J. Math. Anal. Appl.419(2014), no. 2, 738–747. 1297.42019 10.1016/j.jmaa.2014.05.001[9] J. Cunanan and M. Sugimoto,Unimodular Fourier multipliers on Wiener amalgam spaces, J. Math. Anal. Appl.419(2014), no. 2, 738–747. 1297.42019 10.1016/j.jmaa.2014.05.001

10.

[10] H. G. Feichtinger,Banach spaces of distributions defined by decomposition methods, II, Math. Nachr.132(1987), 207–237. 0586.46031 10.1002/mana.19871320116[10] H. G. Feichtinger,Banach spaces of distributions defined by decomposition methods, II, Math. Nachr.132(1987), 207–237. 0586.46031 10.1002/mana.19871320116

11.

[11] H. G. Feichtinger,Modulation spaces on locally compact abelian groups, preprint, http://www.researchgate.net/publication/20052428(accessed 29 August 2017).[11] H. G. Feichtinger,Modulation spaces on locally compact abelian groups, preprint, http://www.researchgate.net/publication/20052428(accessed 29 August 2017).

12.

[12] H. G. Feichtinger and P. Gröbner,Banach spaces of distributions defined by decomposition methods, I, Math. Nachr.123(1985), 97–120.[12] H. G. Feichtinger and P. Gröbner,Banach spaces of distributions defined by decomposition methods, I, Math. Nachr.123(1985), 97–120.

13.

[13] L. Han, B. Wang, and B. Guo,Inviscid limit for the derivative Ginzburg-Landau equation with small data in modulation and Sobolev spaces, Appl. Comput. Harmon. Anal.32(2012), no. 2, 197–222. 1236.35177 10.1016/j.acha.2011.04.001[13] L. Han, B. Wang, and B. Guo,Inviscid limit for the derivative Ginzburg-Landau equation with small data in modulation and Sobolev spaces, Appl. Comput. Harmon. Anal.32(2012), no. 2, 197–222. 1236.35177 10.1016/j.acha.2011.04.001

14.

[14] L. Hörmander,Estimates for translation invariant operators in $L^{p}$ spaces, Acta Math.104(1960), 93–140.[14] L. Hörmander,Estimates for translation invariant operators in $L^{p}$ spaces, Acta Math.104(1960), 93–140.

15.

[15] K. Kato, M. Kobayashi, and S. Ito,Estimates on modulation spaces for Schrödinger evolution operators with quadratic and sub-quadratic potentials, J. Funct. Anal.266(2014), no. 2, 733–753. 1294.35010 10.1016/j.jfa.2013.08.017[15] K. Kato, M. Kobayashi, and S. Ito,Estimates on modulation spaces for Schrödinger evolution operators with quadratic and sub-quadratic potentials, J. Funct. Anal.266(2014), no. 2, 733–753. 1294.35010 10.1016/j.jfa.2013.08.017

16.

[16] A. Miyachi, F. Nicola, S. Rivetti, A. Tabacco, and N. Tomita,Estimates for unimodular Fourier multipliers on modulation spaces, Proc. Amer. Math. Soc.137(2009), no. 11, 3869–3883. 1183.42013 10.1090/S0002-9939-09-09968-7[16] A. Miyachi, F. Nicola, S. Rivetti, A. Tabacco, and N. Tomita,Estimates for unimodular Fourier multipliers on modulation spaces, Proc. Amer. Math. Soc.137(2009), no. 11, 3869–3883. 1183.42013 10.1090/S0002-9939-09-09968-7

17.

[17] F. Nicola,Phase space analysis of semilinear parabolic equations, J. Funct. Anal.267(2014), no. 3, 727–743. 1296.35225 10.1016/j.jfa.2014.05.007[17] F. Nicola,Phase space analysis of semilinear parabolic equations, J. Funct. Anal.267(2014), no. 3, 727–743. 1296.35225 10.1016/j.jfa.2014.05.007

18.

[18] F. Nicola,Convergence in $L^{p}$ for Feynman path integrals, Adv. Math.294(2016), 384–409.[18] F. Nicola,Convergence in $L^{p}$ for Feynman path integrals, Adv. Math.294(2016), 384–409.

19.

[19] M. Ruzhansky, M. Sugimoto, and B. Wang, “Modulation spaces and nonlinear evolution equations” inEvolution Equations of Hyperbolic and Schrödinger Type, Progr. Math.301, Birkhäuser, Basel, 2012, 267–283. 1256.42038[19] M. Ruzhansky, M. Sugimoto, and B. Wang, “Modulation spaces and nonlinear evolution equations” inEvolution Equations of Hyperbolic and Schrödinger Type, Progr. Math.301, Birkhäuser, Basel, 2012, 267–283. 1256.42038

20.

[20] M. Ruzhansky, B. Wang, and H. Zhang,Global well-posedness and scattering for the fourth order nonlinear Schrödinger equations with small data in modulation and Sobolev spaces, J. Math. Pures Appl. (9)105(2016), no. 1, 31–65. 1336.35322[20] M. Ruzhansky, B. Wang, and H. Zhang,Global well-posedness and scattering for the fourth order nonlinear Schrödinger equations with small data in modulation and Sobolev spaces, J. Math. Pures Appl. (9)105(2016), no. 1, 31–65. 1336.35322

21.

[21] M. Sugimoto and N. Tomita,The dilation property of modulation space and their inclusion relation with Besov spaces, J. Funct. Anal.248(2007), no. 1, 79–106. 1124.42018 10.1016/j.jfa.2007.03.015[21] M. Sugimoto and N. Tomita,The dilation property of modulation space and their inclusion relation with Besov spaces, J. Funct. Anal.248(2007), no. 1, 79–106. 1124.42018 10.1016/j.jfa.2007.03.015

22.

[22] N. Tomita,Fractional integrals on modulation spaces, Math. Nachr.279(2006), no. 5–6, 672–680.[22] N. Tomita,Fractional integrals on modulation spaces, Math. Nachr.279(2006), no. 5–6, 672–680.

23.

[23] N. Tomita,On the Hörmander multiplier theorem and modulation spaces, Appl. Comput. Harmon. Anal.26(2009), no. 3, 408–415. 1181.47052 10.1016/j.acha.2008.10.001[23] N. Tomita,On the Hörmander multiplier theorem and modulation spaces, Appl. Comput. Harmon. Anal.26(2009), no. 3, 408–415. 1181.47052 10.1016/j.acha.2008.10.001

24.

[24] N. Tomita, “Unimodular Fourier multipliers on modulation spaces $M_{p,q}$ for $0<p<1$” inHarmonic Analysis and Nonlinear Partial Differential Equations, RIMS Kôkyûroku BessatsuB18, Res. Inst. Math. Sci. (RIMS), Kyoto, 2010, 125–131.[24] N. Tomita, “Unimodular Fourier multipliers on modulation spaces $M_{p,q}$ for $0<p<1$” inHarmonic Analysis and Nonlinear Partial Differential Equations, RIMS Kôkyûroku BessatsuB18, Res. Inst. Math. Sci. (RIMS), Kyoto, 2010, 125–131.

25.

[25] H. Triebel,Theory of Function Spaces, Monogr. Math.78, Birkhäuser, Basel, 1983.[25] H. Triebel,Theory of Function Spaces, Monogr. Math.78, Birkhäuser, Basel, 1983.

26.

[26] B. Wang, Z. Huo, C. Hao, and Z. Guo,Harmonic Analysis Method for Nonlinear Evolution Equations, I, World Scientfic, Hackensack, NJ, 2011.[26] B. Wang, Z. Huo, C. Hao, and Z. Guo,Harmonic Analysis Method for Nonlinear Evolution Equations, I, World Scientfic, Hackensack, NJ, 2011.

27.

[27] G. Zhao, J. Chen, D. Fan, and W. Guo,Sharp estimates of unimodular multipliers on frequency decomposition spaces, Nonlinear Anal.142(2016), 26–47. 1341.42025 10.1016/j.na.2016.04.003[27] G. Zhao, J. Chen, D. Fan, and W. Guo,Sharp estimates of unimodular multipliers on frequency decomposition spaces, Nonlinear Anal.142(2016), 26–47. 1341.42025 10.1016/j.na.2016.04.003
Copyright © 2018 Tusi Mathematical Research Group
Qiang Huang, Jiecheng Chen, Dashan Fan, and Xiangrong Zhu "Hörmander-type theorems on unimodular multipliers and applications to modulation spaces," Banach Journal of Mathematical Analysis 12(1), 85-103, (January 2018). https://doi.org/10.1215/17358787-2017-0037
Received: 27 June 2016; Accepted: 28 January 2017; Published: January 2018
Vol.12 • No. 1 • January 2018
Back to Top