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September  2021, 41(9): 4207-4253. doi: 10.3934/dcds.2021034

Local well-posedness for the derivative nonlinear Schrödinger equation with $ L^2 $-subcritical data

1. 

Department of Mathematics, University of Wisconsin Madison, Madison, WI, USA

2. 

LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China

* Corresponding author: Baoxiang Wang

Received  August 2020 Revised  January 2021 Published  September 2021 Early access  February 2021

Fund Project: The second and third named authors were supported in part by NSFC grant 11771024

Considering the Cauchy problem of the derivative NLS
$ \begin{align} {\rm i} u_{t} + \partial_{xx} u = {\rm i} \mu \partial_x (|u|^2u),\quad u(0,x) = u_0(x), \end{align} $
we will show its local well-posedness in modulation spaces
$ M^{1/2}_{2,q}(\mathbb{R}) $
$ (4{\leqslant} q<\infty) $
. It is well-known that
$ H^{1/2} $
is a critical Sobolev space of the derivative NLS. Noticing that
$ H^{1/2} \subset M^{1/2}_{2,q} \subset B^{1/q}_{2,q} $
$ (q{\geqslant} 2) $
are sharp inclusions, our result contains a class of functions in
$ L^2\setminus H^{1/2} $
.
Citation: Shaoming Guo, Xianfeng Ren, Baoxiang Wang. Local well-posedness for the derivative nonlinear Schrödinger equation with $ L^2 $-subcritical data. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4207-4253. doi: 10.3934/dcds.2021034
References:
[1]

H. Bahouri and G. Perelman, Global well-posedness for the derivative nonlinear Schrödinger equation, Preprint, arXiv: 2012.01923. Google Scholar

[2]

Á. Bényi and K. A. Okoudjou, Local well-posedness of nonlinear dispersive equations on modulation spaces, Bull. Lond. Math. Soc., 41 (2009), 549-558.  doi: 10.1112/blms/bdp027.  Google Scholar

[3]

Á. BényiK. GröchenigK. A. Okoudjou and L. G. Rogers, Unimodular Fourier multipliers for modulation spaces, J. Funct. Anal., 246 (2007), 366-384.  doi: 10.1016/j.jfa.2006.12.019.  Google Scholar

[4]

J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, 1976.  Google Scholar

[5]

H. A. Biagioni and F. Linares, Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations, Trans. Amer. Math. Soc., 353 (2001), 3649-3659.  doi: 10.1090/S0002-9947-01-02754-4.  Google Scholar

[6]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Ⅰ. Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156.  doi: 10.1007/BF01896020.  Google Scholar

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J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Ⅱ. The KdV-equation, Geom. Funct. Anal., 3 (1993), 209-262.  doi: 10.1007/BF01895688.  Google Scholar

[8]

L. ChaichenetsD. HundertmarkP. Kunstmann and N. Pattakos, On the existence of global solutions of the one-dimensional cubic NLS for initial data in the modulation space $M^{p, q}(\mathbb{R})$, J. Differential Equations, 263 (2017), 4429-4441.  doi: 10.1016/j.jde.2017.04.020.  Google Scholar

[9]

L. ChaichenetsD. HundertmarkP. Kunstmann and N. Pattakos, Nonlinear Schrödinger equation, differentiation by parts and modulation spaces, J. Evol. Equ., 19 (2019), 803-843.  doi: 10.1007/s00028-019-00501-z.  Google Scholar

[10]

J. ChenD. Fan and L. Sun, Asymptotic estimates for unimodular Fourier multipliers on modulation spaces, Discrete Contin. Dyn. Syst., 32 (2012), 467-485.  doi: 10.3934/dcds.2012.32.467.  Google Scholar

[11]

M. J. ChenB. X. WangS. X. Wang and M. W. Wong, On dissipative nonlinear evolutional pseudo-differential equations, Appl. Comput. Harmon. Anal., 48 (2020), 182-217.  doi: 10.1016/j.acha.2018.04.003.  Google Scholar

[12]

A. Córdoba and C. Fefferman, Wave packets and Fourier integral operators, Commun. Partial Differ. Equations, 3 (1978), 979-1005.  doi: 10.1080/03605307808820083.  Google Scholar

[13]

E. Cordero and F. Nicola, Metaplectic representation on Wiener amalgam spaces and applications to the Schrödinger equation, J. Funct. Anal., 254 (2008), 506-534.  doi: 10.1016/j.jfa.2007.09.015.  Google Scholar

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E. Cordero and F. Nicola, Some new Strichartz estimates for the Schrödinger equation, J. Differential Equations, 245 (2008), 1945-1974.  doi: 10.1016/j.jde.2008.07.009.  Google Scholar

[15]

E. Cordero and F. Nicola, Remarks on Fourier multipliers and applications to the wave equation, J. Math. Anal. Appl., 353 (2009), 583-591.  doi: 10.1016/j.jmaa.2008.12.027.  Google Scholar

[16]

H. G. Feichtinger, Modulation Spaces on Locally Compact Abelian Group, Technical Report, University of Vienna, 1983. Google Scholar

[17]

K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser, 2001. doi: 10.1007/978-1-4612-0003-1.  Google Scholar

[18]

A. Grünrock, An improved local well-posedness result for the modified KdV equation, Int. Math. Res. Not., 2004 (2004), 3287-3308.  doi: 10.1155/S1073792804140981.  Google Scholar

[19]

B. Guo and Y. P. Wu, Orbital stability of solitary waves for the nonlinear derivative Schrödinger equation, Journal of Differential Equations, 123 (1994), 35-55.  doi: 10.1006/jdeq.1995.1156.  Google Scholar

[20]

A. Grünrock, Bi- and trilinear Schrödinger estimates in one space dimension with applications to cubic NLS and DNLS, Int. Math. Res. Not., 2005 (2005), 2525-2558.  doi: 10.1155/IMRN.2005.2525.  Google Scholar

[21]

S. M. Guo, On the 1D cubic nonlinear Schrödinger equation in an almost critical space, J. Fourier Annl. Appl., 23 (2017), 91-124.  doi: 10.1007/s00041-016-9464-z.  Google Scholar

[22]

Z. Guo and Y. Wu, Global well-posedness for the derivative nonlinear Schrödinger equation in $H^{1/2}$, Discrete and Continuous Dynamical Systems, 37 (2017), 257-264.  doi: 10.3934/dcds.2017010.  Google Scholar

[23]

M. HadacS. Herr and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 917-941.  doi: 10.1016/j.anihpc.2008.04.002.  Google Scholar

[24]

J. S. Han and B. X. Wang, $\alpha$-Modulation spaces (Ⅰ) scaling, embedding and algebraic properties, J. Math. Soc. Japan, 66 (2014), 1315-1373.  doi: 10.2969/jmsj/06641315.  Google Scholar

[25]

N. Hayashi, The initial value problem for the derivative nonlinear Schrödinger equation in the energy space, Nonlinear Anal., 20 (1993), 823-833.  doi: 10.1016/0362-546X(93)90071-Y.  Google Scholar

[26]

N. Hayashi and T. Ozawa, On the derivative nonlinear Schrödinger equation, Phys. D, 55 (1992), 14-36.  doi: 10.1016/0167-2789(92)90185-P.  Google Scholar

[27]

N. Hayashi and T. Ozawa, Finite energy solutions of nonlinear Schrödinger equations of derivative type, SIAM J. Math. Anal., 25 (1994), 1488-1503.  doi: 10.1137/S0036141093246129.  Google Scholar

[28]

N. Hayashi and T. Ozawa, Remarks on nonlinear Schrödinger equations in one space dimension, Differential Integral Equations, 7 (1994), 453-461.   Google Scholar

[29]

T. Iwabuchi, Navier-Stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices, J. Differential Equations, 248 (2010), 1972-2002.  doi: 10.1016/j.jde.2009.08.013.  Google Scholar

[30]

K. KatoM. Kobayashi and S. Ito, Representation on Schrödinger operator of a free partical via short time Fourier transform and its applications, Tohoku Math. J., 64 (2012), 223-231.  doi: 10.2748/tmj/1341249372.  Google Scholar

[31]

K. KatoM. Kobayashi and S. Ito, Estimates on modulation spaces for Schrödinger evolution operators with quadratic and sub-quadratic potentials, J. Funct. Anal., 266 (2014), 733-753.  doi: 10.1016/j.jfa.2013.08.017.  Google Scholar

[32]

T. Kato, The global Cauchy problems for the nonlinear dispersive equations on modulation spaces, J. Math. Anal. Appl., 413 (2014), 821-840.  doi: 10.1016/j.jmaa.2013.12.022.  Google Scholar

[33]

H. Koch and D. Tataru, Dispersive estimates for principlally normal pseudo-differential operators, Comm. Pure Appl. Math., 58 (2005), 217-284.  doi: 10.1002/cpa.20067.  Google Scholar

[34]

H. Koch and D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not., 2007 (2007), Art. ID rnm053, 36 pp. doi: 10.1093/imrn/rnm053.  Google Scholar

[35]

H. Koch and D. Tataru, Energy and local energy bounds for the 1D cubic NLS equation in $H^{1/4}$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 955-988.  doi: 10.1016/j.anihpc.2012.05.006.  Google Scholar

[36]

S. Kwon and Y. Wu, Orbital stability of solitary waves for derivative nonlinear Schrödinger equation, Journal d'Analyse Mathématique, 135 (2018), 473-486.  doi: 10.1007/s11854-018-0038-7.  Google Scholar

[37]

W. MioT. OginoK. Minami and S. Takeda, Modified nonlinear Schrödinger for Alfven waves propagating along the magnetic field in cold plasma, Journal of the Physical Society of Japan, 41 (1976), 265-271.  doi: 10.1143/JPSJ.41.265.  Google Scholar

[38]

E. Mjolhus, On the modulational instability of hydromagnetic waves parallel to the magnetic field, Journal of Plasma Physics, 16 (1976), 321-334.  doi: 10.1017/S0022377800020249.  Google Scholar

[39]

T. Oh and Y. Wang, Global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces, J. Differential Equations, 269 (2020), 612-640.  doi: 10.1016/j.jde.2019.12.017.  Google Scholar

[40]

T. Ozawa and Y. Tsutsumi, Space-time estimates for null gauge forms and nonlinear Schrödinger equations, Differential Integral Equations, 11 (1998), 201-222.   Google Scholar

[41]

M. RuzhanskyB. X. 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., 105 (2016), 31-65.  doi: 10.1016/j.matpur.2015.09.005.  Google Scholar

[42]

M. Sugimoto and N. Tomita, The dilation property of modulation spaces and their inclusion relation with Besov spaces, J. Funct. Anal., 248 (2007), 79-106.  doi: 10.1016/j.jfa.2007.03.015.  Google Scholar

[43]

M. SugimotoB. X. Wang and R. R. Zhang, Local well-posedness for the Davey–Stewartson equation in a generalized Feichtinger algebra, J. Fourier Anal. Appl., 21 (2015), 1105-1129.  doi: 10.1007/s00041-015-9400-7.  Google Scholar

[44]

H. Takaoka, Well-posedness for the one-dimensional nonlinear Schrödinger equationwith the derivative nonlinearity, Adv. Differential Equations, 4 (1999), 561-580.   Google Scholar

[45]

H. Tribel, Theory of Function Spaces, Birkhäuser-Verlag, 1983. doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[46]

J. Toft, Continuity properties for modulation spaces, with applications to pseudo-differential calculus, I, J. Funct. Anal., 207 (2004), 399-429.  doi: 10.1016/j.jfa.2003.10.003.  Google Scholar

[47]

B.X. Wang, Globally well and ill posedness for non-elliptic derivative Schrödinger equations with small rough data, J. Funct. Anal., 265 (2013), 3009-3052.  doi: 10.1016/j.jfa.2013.08.009.  Google Scholar

[48]

B. X. WangL. J. Han and C. Y. Huang, Global smooth effects and well-posedness for the derivative nonlinear Schrödinger equaton with small rough data, Ann. Inst H. Poincare, AN, 26 (2009), 2253-2281.  doi: 10.1016/j.anihpc.2009.03.004.  Google Scholar

[49]

B. X. Wang and C. Y. Huang, Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations, J. Differential Equations, 239 (2007), 213-250.  doi: 10.1016/j.jde.2007.04.009.  Google Scholar

[50]

B. X. Wang, Z. H. Huo, C. C. Hao and Z. H. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations, I, World Scientific Publishing Co., Pte. Ltd., Hackensack, NJ, 2011. doi: 10.1142/9789814360746.  Google Scholar

[51]

B. X. Wang and H. Hudzik, The global Cauchy problem for the NLS and NLKG with small rough data, J. Diff. Eqns., 232 (2007), 36-73.  doi: 10.1016/j.jde.2006.09.004.  Google Scholar

[52]

B. X. WangL. F. Zhao and B. L. Guo, Isometric decomposition operators, function spaces $E^{\lambda}_{p, q}$ and applications to nonlinear evolution operators, J. Funct. Anal., 233 (2006), 1-39.  doi: 10.1016/j.jfa.2005.06.018.  Google Scholar

show all references

References:
[1]

H. Bahouri and G. Perelman, Global well-posedness for the derivative nonlinear Schrödinger equation, Preprint, arXiv: 2012.01923. Google Scholar

[2]

Á. Bényi and K. A. Okoudjou, Local well-posedness of nonlinear dispersive equations on modulation spaces, Bull. Lond. Math. Soc., 41 (2009), 549-558.  doi: 10.1112/blms/bdp027.  Google Scholar

[3]

Á. BényiK. GröchenigK. A. Okoudjou and L. G. Rogers, Unimodular Fourier multipliers for modulation spaces, J. Funct. Anal., 246 (2007), 366-384.  doi: 10.1016/j.jfa.2006.12.019.  Google Scholar

[4]

J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, 1976.  Google Scholar

[5]

H. A. Biagioni and F. Linares, Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations, Trans. Amer. Math. Soc., 353 (2001), 3649-3659.  doi: 10.1090/S0002-9947-01-02754-4.  Google Scholar

[6]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Ⅰ. Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156.  doi: 10.1007/BF01896020.  Google Scholar

[7]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Ⅱ. The KdV-equation, Geom. Funct. Anal., 3 (1993), 209-262.  doi: 10.1007/BF01895688.  Google Scholar

[8]

L. ChaichenetsD. HundertmarkP. Kunstmann and N. Pattakos, On the existence of global solutions of the one-dimensional cubic NLS for initial data in the modulation space $M^{p, q}(\mathbb{R})$, J. Differential Equations, 263 (2017), 4429-4441.  doi: 10.1016/j.jde.2017.04.020.  Google Scholar

[9]

L. ChaichenetsD. HundertmarkP. Kunstmann and N. Pattakos, Nonlinear Schrödinger equation, differentiation by parts and modulation spaces, J. Evol. Equ., 19 (2019), 803-843.  doi: 10.1007/s00028-019-00501-z.  Google Scholar

[10]

J. ChenD. Fan and L. Sun, Asymptotic estimates for unimodular Fourier multipliers on modulation spaces, Discrete Contin. Dyn. Syst., 32 (2012), 467-485.  doi: 10.3934/dcds.2012.32.467.  Google Scholar

[11]

M. J. ChenB. X. WangS. X. Wang and M. W. Wong, On dissipative nonlinear evolutional pseudo-differential equations, Appl. Comput. Harmon. Anal., 48 (2020), 182-217.  doi: 10.1016/j.acha.2018.04.003.  Google Scholar

[12]

A. Córdoba and C. Fefferman, Wave packets and Fourier integral operators, Commun. Partial Differ. Equations, 3 (1978), 979-1005.  doi: 10.1080/03605307808820083.  Google Scholar

[13]

E. Cordero and F. Nicola, Metaplectic representation on Wiener amalgam spaces and applications to the Schrödinger equation, J. Funct. Anal., 254 (2008), 506-534.  doi: 10.1016/j.jfa.2007.09.015.  Google Scholar

[14]

E. Cordero and F. Nicola, Some new Strichartz estimates for the Schrödinger equation, J. Differential Equations, 245 (2008), 1945-1974.  doi: 10.1016/j.jde.2008.07.009.  Google Scholar

[15]

E. Cordero and F. Nicola, Remarks on Fourier multipliers and applications to the wave equation, J. Math. Anal. Appl., 353 (2009), 583-591.  doi: 10.1016/j.jmaa.2008.12.027.  Google Scholar

[16]

H. G. Feichtinger, Modulation Spaces on Locally Compact Abelian Group, Technical Report, University of Vienna, 1983. Google Scholar

[17]

K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser, 2001. doi: 10.1007/978-1-4612-0003-1.  Google Scholar

[18]

A. Grünrock, An improved local well-posedness result for the modified KdV equation, Int. Math. Res. Not., 2004 (2004), 3287-3308.  doi: 10.1155/S1073792804140981.  Google Scholar

[19]

B. Guo and Y. P. Wu, Orbital stability of solitary waves for the nonlinear derivative Schrödinger equation, Journal of Differential Equations, 123 (1994), 35-55.  doi: 10.1006/jdeq.1995.1156.  Google Scholar

[20]

A. Grünrock, Bi- and trilinear Schrödinger estimates in one space dimension with applications to cubic NLS and DNLS, Int. Math. Res. Not., 2005 (2005), 2525-2558.  doi: 10.1155/IMRN.2005.2525.  Google Scholar

[21]

S. M. Guo, On the 1D cubic nonlinear Schrödinger equation in an almost critical space, J. Fourier Annl. Appl., 23 (2017), 91-124.  doi: 10.1007/s00041-016-9464-z.  Google Scholar

[22]

Z. Guo and Y. Wu, Global well-posedness for the derivative nonlinear Schrödinger equation in $H^{1/2}$, Discrete and Continuous Dynamical Systems, 37 (2017), 257-264.  doi: 10.3934/dcds.2017010.  Google Scholar

[23]

M. HadacS. Herr and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 917-941.  doi: 10.1016/j.anihpc.2008.04.002.  Google Scholar

[24]

J. S. Han and B. X. Wang, $\alpha$-Modulation spaces (Ⅰ) scaling, embedding and algebraic properties, J. Math. Soc. Japan, 66 (2014), 1315-1373.  doi: 10.2969/jmsj/06641315.  Google Scholar

[25]

N. Hayashi, The initial value problem for the derivative nonlinear Schrödinger equation in the energy space, Nonlinear Anal., 20 (1993), 823-833.  doi: 10.1016/0362-546X(93)90071-Y.  Google Scholar

[26]

N. Hayashi and T. Ozawa, On the derivative nonlinear Schrödinger equation, Phys. D, 55 (1992), 14-36.  doi: 10.1016/0167-2789(92)90185-P.  Google Scholar

[27]

N. Hayashi and T. Ozawa, Finite energy solutions of nonlinear Schrödinger equations of derivative type, SIAM J. Math. Anal., 25 (1994), 1488-1503.  doi: 10.1137/S0036141093246129.  Google Scholar

[28]

N. Hayashi and T. Ozawa, Remarks on nonlinear Schrödinger equations in one space dimension, Differential Integral Equations, 7 (1994), 453-461.   Google Scholar

[29]

T. Iwabuchi, Navier-Stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices, J. Differential Equations, 248 (2010), 1972-2002.  doi: 10.1016/j.jde.2009.08.013.  Google Scholar

[30]

K. KatoM. Kobayashi and S. Ito, Representation on Schrödinger operator of a free partical via short time Fourier transform and its applications, Tohoku Math. J., 64 (2012), 223-231.  doi: 10.2748/tmj/1341249372.  Google Scholar

[31]

K. KatoM. Kobayashi and S. Ito, Estimates on modulation spaces for Schrödinger evolution operators with quadratic and sub-quadratic potentials, J. Funct. Anal., 266 (2014), 733-753.  doi: 10.1016/j.jfa.2013.08.017.  Google Scholar

[32]

T. Kato, The global Cauchy problems for the nonlinear dispersive equations on modulation spaces, J. Math. Anal. Appl., 413 (2014), 821-840.  doi: 10.1016/j.jmaa.2013.12.022.  Google Scholar

[33]

H. Koch and D. Tataru, Dispersive estimates for principlally normal pseudo-differential operators, Comm. Pure Appl. Math., 58 (2005), 217-284.  doi: 10.1002/cpa.20067.  Google Scholar

[34]

H. Koch and D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not., 2007 (2007), Art. ID rnm053, 36 pp. doi: 10.1093/imrn/rnm053.  Google Scholar

[35]

H. Koch and D. Tataru, Energy and local energy bounds for the 1D cubic NLS equation in $H^{1/4}$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 955-988.  doi: 10.1016/j.anihpc.2012.05.006.  Google Scholar

[36]

S. Kwon and Y. Wu, Orbital stability of solitary waves for derivative nonlinear Schrödinger equation, Journal d'Analyse Mathématique, 135 (2018), 473-486.  doi: 10.1007/s11854-018-0038-7.  Google Scholar

[37]

W. MioT. OginoK. Minami and S. Takeda, Modified nonlinear Schrödinger for Alfven waves propagating along the magnetic field in cold plasma, Journal of the Physical Society of Japan, 41 (1976), 265-271.  doi: 10.1143/JPSJ.41.265.  Google Scholar

[38]

E. Mjolhus, On the modulational instability of hydromagnetic waves parallel to the magnetic field, Journal of Plasma Physics, 16 (1976), 321-334.  doi: 10.1017/S0022377800020249.  Google Scholar

[39]

T. Oh and Y. Wang, Global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces, J. Differential Equations, 269 (2020), 612-640.  doi: 10.1016/j.jde.2019.12.017.  Google Scholar

[40]

T. Ozawa and Y. Tsutsumi, Space-time estimates for null gauge forms and nonlinear Schrödinger equations, Differential Integral Equations, 11 (1998), 201-222.   Google Scholar

[41]

M. RuzhanskyB. X. 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., 105 (2016), 31-65.  doi: 10.1016/j.matpur.2015.09.005.  Google Scholar

[42]

M. Sugimoto and N. Tomita, The dilation property of modulation spaces and their inclusion relation with Besov spaces, J. Funct. Anal., 248 (2007), 79-106.  doi: 10.1016/j.jfa.2007.03.015.  Google Scholar

[43]

M. SugimotoB. X. Wang and R. R. Zhang, Local well-posedness for the Davey–Stewartson equation in a generalized Feichtinger algebra, J. Fourier Anal. Appl., 21 (2015), 1105-1129.  doi: 10.1007/s00041-015-9400-7.  Google Scholar

[44]

H. Takaoka, Well-posedness for the one-dimensional nonlinear Schrödinger equationwith the derivative nonlinearity, Adv. Differential Equations, 4 (1999), 561-580.   Google Scholar

[45]

H. Tribel, Theory of Function Spaces, Birkhäuser-Verlag, 1983. doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[46]

J. Toft, Continuity properties for modulation spaces, with applications to pseudo-differential calculus, I, J. Funct. Anal., 207 (2004), 399-429.  doi: 10.1016/j.jfa.2003.10.003.  Google Scholar

[47]

B.X. Wang, Globally well and ill posedness for non-elliptic derivative Schrödinger equations with small rough data, J. Funct. Anal., 265 (2013), 3009-3052.  doi: 10.1016/j.jfa.2013.08.009.  Google Scholar

[48]

B. X. WangL. J. Han and C. Y. Huang, Global smooth effects and well-posedness for the derivative nonlinear Schrödinger equaton with small rough data, Ann. Inst H. Poincare, AN, 26 (2009), 2253-2281.  doi: 10.1016/j.anihpc.2009.03.004.  Google Scholar

[49]

B. X. Wang and C. Y. Huang, Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations, J. Differential Equations, 239 (2007), 213-250.  doi: 10.1016/j.jde.2007.04.009.  Google Scholar

[50]

B. X. Wang, Z. H. Huo, C. C. Hao and Z. H. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations, I, World Scientific Publishing Co., Pte. Ltd., Hackensack, NJ, 2011. doi: 10.1142/9789814360746.  Google Scholar

[51]

B. X. Wang and H. Hudzik, The global Cauchy problem for the NLS and NLKG with small rough data, J. Diff. Eqns., 232 (2007), 36-73.  doi: 10.1016/j.jde.2006.09.004.  Google Scholar

[52]

B. X. WangL. F. Zhao and B. L. Guo, Isometric decomposition operators, function spaces $E^{\lambda}_{p, q}$ and applications to nonlinear evolution operators, J. Funct. Anal., 233 (2006), 1-39.  doi: 10.1016/j.jfa.2005.06.018.  Google Scholar

Table 1.  In $ \lambda_1,...,\lambda_3 $, there is at least one frequency near $ \lambda_0 $
$ {\rm Case} $ $ \lambda_1\in $ $ \lambda_2\in $ $ \lambda_3\in $
$ h_-h_-a $ $ [\lambda_0, 3\lambda_0/4) $ $ [\lambda_0, 3\lambda_0/4) $ $ [\lambda_0, \infty) $
$ h_-l_-l_- $ $ [\lambda_0, 3\lambda_0/4) $ $ [3\lambda_0/4, \ 0) $ $ [\lambda_0, 0) $
$ h_-l_- a_+ $ $ [\lambda_0, 3\lambda_0/4) $ $ [3\lambda_0/4,0) $ $ [0, \infty) $
$ h_-a_+ a $ $ [\lambda_0, 3\lambda_0/4) $ $ [0, \infty) $ $ [\lambda_0, \infty) $
$ {\rm Case} $ $ \lambda_1\in $ $ \lambda_2\in $ $ \lambda_3\in $
$ h_-h_-a $ $ [\lambda_0, 3\lambda_0/4) $ $ [\lambda_0, 3\lambda_0/4) $ $ [\lambda_0, \infty) $
$ h_-l_-l_- $ $ [\lambda_0, 3\lambda_0/4) $ $ [3\lambda_0/4, \ 0) $ $ [\lambda_0, 0) $
$ h_-l_- a_+ $ $ [\lambda_0, 3\lambda_0/4) $ $ [3\lambda_0/4,0) $ $ [0, \infty) $
$ h_-a_+ a $ $ [\lambda_0, 3\lambda_0/4) $ $ [0, \infty) $ $ [\lambda_0, \infty) $
Table 2.  $ \lambda_1,\lambda_2,\lambda_3 $ far away from $ \lambda_0 $
$ {\rm Case} $ $ \lambda_1\in $ $ \lambda_2\in $ $ \lambda_3\in $
$ l_-l_-l $ $ [3\lambda_0/4, 0) $ $ [3\lambda_0/4,0) $ $ [\lambda_0, -\lambda_0/16) $
$ l_-l_-h_+ $ $ [ 3\lambda_0/4, 0) $ $ [3\lambda_0/4, \ 0) $ $ [-\lambda_0/16, \infty) $
$ l_-a_+a_m $ $ [3\lambda_0/4, 0) $ $ [0, \infty) $ $ [\lambda_0, -\lambda_0) $
$ l_-a_+h_+ $ $ [3\lambda_0/4, 0) $ $ [0, \infty) $ $ [-\lambda_0, \infty) $
$ {\rm Case} $ $ \lambda_1\in $ $ \lambda_2\in $ $ \lambda_3\in $
$ l_-l_-l $ $ [3\lambda_0/4, 0) $ $ [3\lambda_0/4,0) $ $ [\lambda_0, -\lambda_0/16) $
$ l_-l_-h_+ $ $ [ 3\lambda_0/4, 0) $ $ [3\lambda_0/4, \ 0) $ $ [-\lambda_0/16, \infty) $
$ l_-a_+a_m $ $ [3\lambda_0/4, 0) $ $ [0, \infty) $ $ [\lambda_0, -\lambda_0) $
$ l_-a_+h_+ $ $ [3\lambda_0/4, 0) $ $ [0, \infty) $ $ [-\lambda_0, \infty) $
Table 3.  $ \lambda_1,\lambda_2,\lambda_3 {\geqslant} 0 $
$ {\rm Case} $ $ \lambda_1\in $ $ \lambda_2\in $ $ \lambda_3\in $
$ a_+a_+h $ $ [0, \infty) $ $ [0, \infty) $ $ [\lambda_0, \infty) $
$ {\rm Case} $ $ \lambda_1\in $ $ \lambda_2\in $ $ \lambda_3\in $
$ a_+a_+h $ $ [0, \infty) $ $ [0, \infty) $ $ [\lambda_0, \infty) $
Table 4.  $ \lambda_0 \ll 0 $, $ \lambda_1<\lambda_0 {\leqslant} \lambda_2\wedge \lambda_3 $
$ {\rm Case} $ $ \lambda_1\in $ $ \lambda_2\in $ $ \lambda_3\in $
$ 1 $ $ [5\lambda_0/4-C, \lambda_0) $ $ [\lambda_0, 3\lambda_0/4) $ $ [\lambda_0, 3\lambda_0/4 +C) $
$ 2 $ $ [2\lambda_0 -C, \lambda_0) $ $ [3\lambda_0/4, \ 0) $ $ [\lambda_0, C) $
$ 3 $ $ (11\lambda_0/4-C, \lambda_0) $ $ [0, -3\lambda_0/4) $ $ [\lambda_0, -3\lambda_0/4 +C) $
$ 4 $ $ (-\infty, \lambda_0) $ $ [-3\lambda_0/4, \infty) $ $ [\lambda_0, \infty) $
$ {\rm Case} $ $ \lambda_1\in $ $ \lambda_2\in $ $ \lambda_3\in $
$ 1 $ $ [5\lambda_0/4-C, \lambda_0) $ $ [\lambda_0, 3\lambda_0/4) $ $ [\lambda_0, 3\lambda_0/4 +C) $
$ 2 $ $ [2\lambda_0 -C, \lambda_0) $ $ [3\lambda_0/4, \ 0) $ $ [\lambda_0, C) $
$ 3 $ $ (11\lambda_0/4-C, \lambda_0) $ $ [0, -3\lambda_0/4) $ $ [\lambda_0, -3\lambda_0/4 +C) $
$ 4 $ $ (-\infty, \lambda_0) $ $ [-3\lambda_0/4, \infty) $ $ [\lambda_0, \infty) $
Table 5.  Two subcases of Case 4
$ {\rm Case} $ $ \lambda_1\in $ $ \lambda_2\in $ $ \lambda_3\in $
$ 4.1 $ $ (-\infty, \lambda_0) $ $ [-3\lambda_0/4, \infty) $ $ [\lambda_0, 0) $
$ 4.2 $ $ (-\infty, \lambda_0) $ $ [-3\lambda_0/4, \infty) $ $ [0, \infty) $
$ {\rm Case} $ $ \lambda_1\in $ $ \lambda_2\in $ $ \lambda_3\in $
$ 4.1 $ $ (-\infty, \lambda_0) $ $ [-3\lambda_0/4, \infty) $ $ [\lambda_0, 0) $
$ 4.2 $ $ (-\infty, \lambda_0) $ $ [-3\lambda_0/4, \infty) $ $ [0, \infty) $
Table 6.  $ \lambda_0{\geqslant}\lambda_1{\geqslant}\lambda_3{\geqslant} \lambda_2 {\geqslant} \lambda_4{\geqslant} \lambda_5 $, only one higher frequency in $ \lambda_1,...,\lambda_5 $
$ {\rm Case} $ $ \lambda_1\in $ $ \lambda_3\in $ $ \lambda_2\in $ $ \lambda_4\in $ $ \lambda_5\in $
$ hllll $ $ h $ $ l $ $ l $ $ l $ $ l $
$ hllll_- $ $ h $ $ l $ $ l $ $ l $ $ l_- $
$ hlll_-l_- $ $ h $ $ l $ $ l $ $ l_- $ $ l_- $
$ hll_-l_-l_- $ $ h $ $ l $ $ l_- $ $ l_- $ $ l_- $
$ hl_-l_-l_-l_- $ $ h $ $ l_- $ $ l_- $ $ l_- $ $ l_- $
$ {\rm Case} $ $ \lambda_1\in $ $ \lambda_3\in $ $ \lambda_2\in $ $ \lambda_4\in $ $ \lambda_5\in $
$ hllll $ $ h $ $ l $ $ l $ $ l $ $ l $
$ hllll_- $ $ h $ $ l $ $ l $ $ l $ $ l_- $
$ hlll_-l_- $ $ h $ $ l $ $ l $ $ l_- $ $ l_- $
$ hll_-l_-l_- $ $ h $ $ l $ $ l_- $ $ l_- $ $ l_- $
$ hl_-l_-l_-l_- $ $ h $ $ l_- $ $ l_- $ $ l_- $ $ l_- $
Table 7.  $ \lambda_0{\geqslant}\lambda_2{\geqslant}\lambda_1{\geqslant} \lambda_3 {\geqslant} \lambda_5{\geqslant} \lambda_4 $, only one higher frequency in $ \lambda_1,...,\lambda_5 $
$ {\rm Case} $ $ \lambda_2\in $ $ \lambda_1\in $ $ \lambda_3\in $ $ \lambda_5\in $ $ \lambda_4\in $
$ llllh_- $ $ l $ $ l $ $ l $ $ l $ $ h_- $
$ llll_-h_- $ $ l $ $ l $ $ l $ $ l_- $ $ h_- $
$ hlll_-l_-h_- $ $ l $ $ l $ $ l_- $ $ l_- $ $ h_- $
$ ll_-l_-l_-h_- $ $ l $ $ l_- $ $ l_- $ $ l_- $ $ h_- $
$ l_-l_-l_-l_-h_- $ $ l_- $ $ l_- $ $ l_- $ $ l_- $ $ h_- $
$ {\rm Case} $ $ \lambda_2\in $ $ \lambda_1\in $ $ \lambda_3\in $ $ \lambda_5\in $ $ \lambda_4\in $
$ llllh_- $ $ l $ $ l $ $ l $ $ l $ $ h_- $
$ llll_-h_- $ $ l $ $ l $ $ l $ $ l_- $ $ h_- $
$ hlll_-l_-h_- $ $ l $ $ l $ $ l_- $ $ l_- $ $ h_- $
$ ll_-l_-l_-h_- $ $ l $ $ l_- $ $ l_- $ $ l_- $ $ h_- $
$ l_-l_-l_-l_-h_- $ $ l_- $ $ l_- $ $ l_- $ $ l_- $ $ h_- $
Table 8.  $ \lambda_0{\geqslant}\lambda_1{\geqslant}\lambda_2{\geqslant} \lambda_3 {\geqslant} \lambda_5{\geqslant} \lambda_4 $, only one higher frequency in $ \lambda_1,...,\lambda_5 $
$ {\rm Case} $ $ \lambda_1\in $ $ \lambda_2\in $ $ \lambda_3\in $ $ \lambda_5\in $ $ \lambda_4\in $
$ hllll $ $ h $ $ l $ $ l $ $ l $ $ l $
$ hllll_- $ $ h $ $ l $ $ l $ $ l $ $ l_- $
$ hlll_-l_- $ $ h $ $ l $ $ l $ $ l_- $ $ l_- $
$ hll_-l_-l_- $ $ h $ $ l $ $ l_- $ $ l_- $ $ l_- $
$ hl_-l_-l_-l_- $ $ h $ $ l_- $ $ l_- $ $ l_- $ $ l_- $
$ {\rm Case} $ $ \lambda_1\in $ $ \lambda_2\in $ $ \lambda_3\in $ $ \lambda_5\in $ $ \lambda_4\in $
$ hllll $ $ h $ $ l $ $ l $ $ l $ $ l $
$ hllll_- $ $ h $ $ l $ $ l $ $ l $ $ l_- $
$ hlll_-l_- $ $ h $ $ l $ $ l $ $ l_- $ $ l_- $
$ hll_-l_-l_- $ $ h $ $ l $ $ l_- $ $ l_- $ $ l_- $
$ hl_-l_-l_-l_- $ $ h $ $ l_- $ $ l_- $ $ l_- $ $ l_- $
Table 9.  $ \lambda_0{\geqslant}\lambda_1{\geqslant}\lambda_2{\geqslant} \lambda_3 {\geqslant} \lambda_5{\geqslant} \lambda_4 $, only one higher frequency in $ \lambda_1,...,\lambda_5 $
$ {\rm Case} $ $ \lambda_1\in $ $ \lambda_2\in $ $ \lambda_3\in $ $ \lambda_5\in $ $ \lambda_4\in $
$ llllh_- $ $ l $ $ l $ $ l $ $ l $ $ h_- $
$ llll_-h_- $ $ l $ $ l $ $ l $ $ l_- $ $ h_- $
$ hlll_-l_-h_- $ $ l $ $ l $ $ l_- $ $ l_- $ $ h_- $
$ ll_-l_-l_-h_- $ $ l $ $ l_- $ $ l_- $ $ l_- $ $ h_- $
$ l_-l_-l_-l_-h_- $ $ l_- $ $ l_- $ $ l_- $ $ l_- $ $ h_- $
$ {\rm Case} $ $ \lambda_1\in $ $ \lambda_2\in $ $ \lambda_3\in $ $ \lambda_5\in $ $ \lambda_4\in $
$ llllh_- $ $ l $ $ l $ $ l $ $ l $ $ h_- $
$ llll_-h_- $ $ l $ $ l $ $ l $ $ l_- $ $ h_- $
$ hlll_-l_-h_- $ $ l $ $ l $ $ l_- $ $ l_- $ $ h_- $
$ ll_-l_-l_-h_- $ $ l $ $ l_- $ $ l_- $ $ l_- $ $ h_- $
$ l_-l_-l_-l_-h_- $ $ l_- $ $ l_- $ $ l_- $ $ l_- $ $ h_- $
Table 10.  $ \lambda_0{\geqslant}\lambda_1{\geqslant}\lambda_3{\geqslant} \lambda_5 {\geqslant} \lambda_2{\geqslant} \lambda_4 $, only one higher frequency in $ \lambda_1,...,\lambda_5 $
$ {\rm Case} $ $ \lambda_1\in $ $ \lambda_3\in $ $ \lambda_5\in $ $ \lambda_2\in $ $ \lambda_4\in $
$ hllll $ $ h $ $ l $ $ l $ $ l $ $ l $
$ hllll_- $ $ h $ $ l $ $ l $ $ l $ $ l_- $
$ hlll_-l_- $ $ h $ $ l $ $ l $ $ l_- $ $ l_- $
$ hll_-l_-l_- $ $ h $ $ l $ $ l_- $ $ l_- $ $ l_- $
$ hl_-l_-l_-l_- $ $ h $ $ l_- $ $ l_- $ $ l_- $ $ l_- $
$ {\rm Case} $ $ \lambda_1\in $ $ \lambda_3\in $ $ \lambda_5\in $ $ \lambda_2\in $ $ \lambda_4\in $
$ hllll $ $ h $ $ l $ $ l $ $ l $ $ l $
$ hllll_- $ $ h $ $ l $ $ l $ $ l $ $ l_- $
$ hlll_-l_- $ $ h $ $ l $ $ l $ $ l_- $ $ l_- $
$ hll_-l_-l_- $ $ h $ $ l $ $ l_- $ $ l_- $ $ l_- $
$ hl_-l_-l_-l_- $ $ h $ $ l_- $ $ l_- $ $ l_- $ $ l_- $
Table 11.  $ \lambda_0{\geqslant}\lambda_1{\geqslant}\lambda_3{\geqslant} \lambda_5 {\geqslant} \lambda_2{\geqslant} \lambda_4 $, only one higher frequency in $ \lambda_1,...,\lambda_5 $
$ {\rm Case} $ $ \lambda_1\in $ $ \lambda_3\in $ $ \lambda_5\in $ $ \lambda_2\in $ $ \lambda_4\in $
$ llllh_- $ $ l $ $ l $ $ l $ $ l $ $ h_- $
$ llll_-h_- $ $ l $ $ l $ $ l $ $ l_- $ $ h_- $
$ hlll_-l_-h_- $ $ l $ $ l $ $ l_- $ $ l_- $ $ h_- $
$ ll_-l_-l_-h_- $ $ l $ $ l_- $ $ l_- $ $ l_- $ $ h_- $
$ l_-l_-l_-l_-h_- $ $ l_- $ $ l_- $ $ l_- $ $ l_- $ $ h_- $
$ {\rm Case} $ $ \lambda_1\in $ $ \lambda_3\in $ $ \lambda_5\in $ $ \lambda_2\in $ $ \lambda_4\in $
$ llllh_- $ $ l $ $ l $ $ l $ $ l $ $ h_- $
$ llll_-h_- $ $ l $ $ l $ $ l $ $ l_- $ $ h_- $
$ hlll_-l_-h_- $ $ l $ $ l $ $ l_- $ $ l_- $ $ h_- $
$ ll_-l_-l_-h_- $ $ l $ $ l_- $ $ l_- $ $ l_- $ $ h_- $
$ l_-l_-l_-l_-h_- $ $ l_- $ $ l_- $ $ l_- $ $ l_- $ $ h_- $
Table 12.  $ \lambda_0{\geqslant}\lambda_1{\geqslant}\lambda_3{\geqslant} \lambda_2 {\geqslant} \lambda_5 {\geqslant} \lambda_4 $, only one higher frequency in $ \lambda_1,...,\lambda_5 $
$ {\rm Case} $ $ \lambda_1\in $ $ \lambda_3\in $ $ \lambda_2\in $ $ \lambda_5\in $ $ \lambda_4\in $
$ hllll $ $ h $ $ l $ $ l $ $ l $ $ l $
$ hllll_- $ $ h $ $ l $ $ l $ $ l $ $ l_- $
$ hlll_-l_- $ $ h $ $ l $ $ l $ $ l_- $ $ l_- $
$ hll_-l_-l_- $ $ h $ $ l $ $ l_- $ $ l_- $ $ l_- $
$ hl_-l_-l_-l_- $ $ h $ $ l_- $ $ l_- $ $ l_- $ $ l_- $
$ {\rm Case} $ $ \lambda_1\in $ $ \lambda_3\in $ $ \lambda_2\in $ $ \lambda_5\in $ $ \lambda_4\in $
$ hllll $ $ h $ $ l $ $ l $ $ l $ $ l $
$ hllll_- $ $ h $ $ l $ $ l $ $ l $ $ l_- $
$ hlll_-l_- $ $ h $ $ l $ $ l $ $ l_- $ $ l_- $
$ hll_-l_-l_- $ $ h $ $ l $ $ l_- $ $ l_- $ $ l_- $
$ hl_-l_-l_-l_- $ $ h $ $ l_- $ $ l_- $ $ l_- $ $ l_- $
Table 13.  $ \lambda_0{\geqslant}\lambda_1{\geqslant}\lambda_3{\geqslant} \lambda_2 {\geqslant} \lambda_5 {\geqslant} \lambda_4 $, only one higher frequency in $ \lambda_1,...,\lambda_5 $
$ {\rm Case} $ $ \lambda_1\in $ $ \lambda_3\in $ $ \lambda_2\in $ $ \lambda_5\in $ $ \lambda_4\in $
$ llllh_- $ $ l $ $ l $ $ l $ $ l $ $ h_- $
$ llll_-h_- $ $ l $ $ l $ $ l $ $ l_- $ $ h_- $
$ hlll_-l_-h_- $ $ l $ $ l $ $ l_- $ $ l_- $ $ h_- $
$ ll_-l_-l_-h_- $ $ l $ $ l_- $ $ l_- $ $ l_- $ $ h_- $
$ l_-l_-l_-l_-h_- $ $ l_- $ $ l_- $ $ l_- $ $ l_- $ $ h_- $
$ {\rm Case} $ $ \lambda_1\in $ $ \lambda_3\in $ $ \lambda_2\in $ $ \lambda_5\in $ $ \lambda_4\in $
$ llllh_- $ $ l $ $ l $ $ l $ $ l $ $ h_- $
$ llll_-h_- $ $ l $ $ l $ $ l $ $ l_- $ $ h_- $
$ hlll_-l_-h_- $ $ l $ $ l $ $ l_- $ $ l_- $ $ h_- $
$ ll_-l_-l_-h_- $ $ l $ $ l_- $ $ l_- $ $ l_- $ $ h_- $
$ l_-l_-l_-l_-h_- $ $ l_- $ $ l_- $ $ l_- $ $ l_- $ $ h_- $
Table 14.  $ \lambda_1{\geqslant}\lambda_0{\geqslant}\lambda_2{\geqslant} \lambda_3 {\geqslant} \lambda_4 {\geqslant} \lambda_5 $, only one higher frequency in $ \lambda_1,...,\lambda_5 $
$ {\rm Case} $ $ \lambda_1\in $ $ \lambda_2\in $ $ \lambda_3\in $ $ \lambda_4\in $ $ \lambda_5\in $
$ 2hllll $ $ [\lambda_0, 20\lambda_0] $ $ l $ $ l $ $ l $ $ l $
$ 2hllll_- $ $ [\lambda_0, 20\lambda_0] $ $ l $ $ l $ $ l $ $ l_- $
$ 2hlll_-l_- $ $ [\lambda_0, 20\lambda_0] $ $ l $ $ l $ $ l_- $ $ l_- $
$ 2hll_-l_-l_- $ $ [\lambda_0, 20\lambda_0] $ $ l $ $ l_- $ $ l_- $ $ l_- $
$ 2hl_-l_-l_-l_- $ $ [\lambda_0, 20\lambda_0] $ $ l_- $ $ l_- $ $ l_- $ $ l_- $
$ {\rm Case} $ $ \lambda_1\in $ $ \lambda_2\in $ $ \lambda_3\in $ $ \lambda_4\in $ $ \lambda_5\in $
$ 2hllll $ $ [\lambda_0, 20\lambda_0] $ $ l $ $ l $ $ l $ $ l $
$ 2hllll_- $ $ [\lambda_0, 20\lambda_0] $ $ l $ $ l $ $ l $ $ l_- $
$ 2hlll_-l_- $ $ [\lambda_0, 20\lambda_0] $ $ l $ $ l $ $ l_- $ $ l_- $
$ 2hll_-l_-l_- $ $ [\lambda_0, 20\lambda_0] $ $ l $ $ l_- $ $ l_- $ $ l_- $
$ 2hl_-l_-l_-l_- $ $ [\lambda_0, 20\lambda_0] $ $ l_- $ $ l_- $ $ l_- $ $ l_- $
Table 15.  $ \lambda_0{\geqslant}\lambda_1{\geqslant}\lambda_2{\geqslant} \lambda_3 {\geqslant} \lambda_5 {\geqslant} \lambda_4 $, only one higher frequency in $ \lambda_1,...,\lambda_5 $
$ {\rm Case} $ $ \lambda_1\in $ $ \lambda_2\in $ $ \lambda_3\in $ $ \lambda_5\in $ $ \lambda_4\in $
$ 3llllh_- $ $ l $ $ l $ $ l $ $ l $ $ [-20\lambda_0,-\lambda_0] $
$ 3llll_-h_- $ $ l $ $ l $ $ l $ $ l_- $ $ [-20\lambda_0,-\lambda_0] $
$ 3lll_-l_-h_- $ $ l $ $ l $ $ l_- $ $ l_- $ $ [-20\lambda_0,-\lambda_0] $
$ 3ll_-l_-l_-h_- $ $ l $ $ l_- $ $ l_- $ $ l_- $ $ [-20\lambda_0,-\lambda_0] $
$ 2l_-l_-l_-l_-h_- $ $ l_- $ $ l_- $ $ l_- $ $ l_- $ $ [-20\lambda_0,-\lambda_0] $
$ {\rm Case} $ $ \lambda_1\in $ $ \lambda_2\in $ $ \lambda_3\in $ $ \lambda_5\in $ $ \lambda_4\in $
$ 3llllh_- $ $ l $ $ l $ $ l $ $ l $ $ [-20\lambda_0,-\lambda_0] $
$ 3llll_-h_- $ $ l $ $ l $ $ l $ $ l_- $ $ [-20\lambda_0,-\lambda_0] $
$ 3lll_-l_-h_- $ $ l $ $ l $ $ l_- $ $ l_- $ $ [-20\lambda_0,-\lambda_0] $
$ 3ll_-l_-l_-h_- $ $ l $ $ l_- $ $ l_- $ $ l_- $ $ [-20\lambda_0,-\lambda_0] $
$ 2l_-l_-l_-l_-h_- $ $ l_- $ $ l_- $ $ l_- $ $ l_- $ $ [-20\lambda_0,-\lambda_0] $
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