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On a class of linearly coupled systems on $ \mathbb{R}^N $ involving asymptotically linear terms
Molecular decomposition and a class of Fourier multipliers for bi-parameter modulation spaces
Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China |
In this paper, we investigate bi-parameter modulation spaces on the product of two Euclidean spaces $ \mathbb{R}^{n} $ and $ \mathbb{R}^{m} $ via uniform decompositions of each factor. A molecular decomposition of these bi-parameter spaces are given, which generalizes the related single-parameter result of Kobayashi and Sawano [
References:
[1] |
R. Balan, P. G. Casazza, C. Heil and Z. Landau,
Density, overcompleteness, and localization of frames, Ⅱ, Gabor systems, J. Fourier Anal. Appl., 12 (2006), 309-344.
doi: 10.1007/s00041-005-5035-4. |
[2] |
Á. Bényi, L. Grafakos, K. Gröchenig and K. Okoudjou,
A class of Fourier multipliers for modulation spaces, Appl. Comput. Harmon. Anal., 19 (2005), 131-139.
doi: 10.1016/j.acha.2005.02.002. |
[3] |
Á. Bényi and K. Okoudjou,
Local well-posedness of nonlinear dispersive equations on modulation spaces, Bull. Lond. Math. Soc., 41 (2009), 549-558.
doi: 10.1112/blms/bdp027. |
[4] |
S-Y. A. Chang and R. Fefferman,
Some recent developments in Fourier analysis and $ H^{p} $ theory on product domains, Bull. Amer. Math. Soc., 12 (1985), 1-43.
doi: 10.1090/S0273-0979-1985-15291-7. |
[5] |
S-Y. A. Chang and R. Fefferman,
The Calderón-Zygmund decomposition on product domains, Amer. J. Math., 104 (1982), 455-468.
doi: 10.2307/2374150. |
[6] |
S-Y. A. Chang and R. Fefferman,
A continuous version of duality of $H^{1}$ with $BMO$ on the bidisc, Ann. of Math., 112 (1980), 179-201.
doi: 10.2307/1971324. |
[7] |
J. Chen,
Hörmander type theorem for Fourier multipliers with optimal smoothness on Hardy spaces of arbitrary number of parameters, Acta Math. Sin. (Engl. Ser.), 33 (2017), 1083-1106.
doi: 10.1007/s10114-017-6526-3. |
[8] |
J. Chen and G. Lu,
Hörmander type theorems for multi-linear and multi-parameter Fourier multiplier operators with limited smoothness, Nonlinear Anal., 101 (2014), 98-112.
doi: 10.1016/j.na.2014.01.005. |
[9] |
J. Chen and G. Lu,
Hörmander type theorem on bi-parameter Hardy spaces for bi-parameter Fourier multipliers with optimal smoothness, Rev. Mat. Iberoam., 34 (2018), 1541-1561.
doi: 10.4171/rmi/1035. |
[10] |
W. Ding and G. Lu,
Duality of multi-parameter Triebel-Lizorkin spaces associated with the composition of two singular integral operators, Tans. Amer. Math. Soc., 368 (2016), 7119-7152.
doi: 10.1090/tran/6576. |
[11] |
Y. Ding, G. Lu and B. Ma,
Multi-parameter Triebel-Lizorkin and Besov spaces associated with flag singular integrals, Acta Math. Sin. (Engl. Ser.), 26 (2010), 603-620.
doi: 10.1007/s10114-010-8352-8. |
[12] |
R. Fefferman,
Harmonic Analysis on product spaces, Ann. of Math., 126 (1987), 109-130.
doi: 10.2307/1971346. |
[13] |
R. Fefferman and J. Pipher,
Multiparameter operators and sharp weighted inequalities, Amer. J. Math., 119 (1997), 337-369.
doi: 10.1353/ajm.1997.0011. |
[14] |
R. Fefferman and E. M. Stein,
Singular integrals on product spaces, Adv. Math., 45 (1982), 117-143.
doi: 10.1016/S0001-8708(82)80001-7. |
[15] |
H. G. Feichtinger, Modulation spaces on locally compact abelian groups, Technical report, University Vienna, January 1983. |
[16] |
G. Feichtinger and K. Gröchenig,
Gabor frames and time-frequency analysis of distributions, J. Funct. Anal., 146 (1997), 464-495.
doi: 10.1006/jfan.1996.3078. |
[17] |
H. G. Feichtinger and G. Narimani,
Fourier multipliers of classical modulation spaces, Appl. Comput. Harmon. Anal., 21 (2006), 349-359.
doi: 10.1016/j.acha.2006.04.010. |
[18] |
S. H. Ferguson and M. T. Lacey,
A characterization of product BMO by commutators, Acta Math., 189 (2002), 143-160.
doi: 10.1007/BF02392840. |
[19] |
K. Gröchenig, Foundations of Time-frequency Analysis, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, 2001.
doi: 10.1007/BF02392840. |
[20] |
K. Gröchenig and C. Heil,
Modulation spaces and pseudodifferential operators, Integral Equations Operator Theory, 34 (1999), 439-457.
doi: 10.1007/BF01272884. |
[21] |
K. Gröchenig and Z. Rzeszotnik,
Banach algebras of pseudodifferential operators and their almost diagonalization, Ann. Inst. Fourier (Grenoble), 58 (2008), 2279-2314.
doi: 10.5802/aif.2414. |
[22] |
R. Gundy and E. M. Stein,
$ H^{p} $ theory for the polydisk, Proc. Nat. Acad. Sci., 76 (1979), 1026-1029.
doi: 10.1073/pnas.76.3.1026. |
[23] |
Y. Han, J. Li and G. Lu,
Multiparameter Hardy space theory on Carnot-Carathéodory spaces and product spaces of homogeneous type, Trans. Amer. Math. Soc., 365 (2013), 319-360.
doi: 10.1090/S0002-9947-2012-05638-8. |
[24] |
Y. Han and G. Lu, Discrete Littlewood-Paley-Stein theory and multi-parameter Hardy spaces associated with the flag singular integrals, preprint, arXiv: 0801.1701.
doi: 10.1090/S0002-9947-2012-05638-8. |
[25] |
Y. Han, G. Lu and Z. Ruan,
Boundedness criterion of Journé's class of singular integrals on multiparameter Hardy spaces, J. Funct. Anal., 264 (2013), 1238-1268.
doi: 10.1016/j.jfa.2012.12.006. |
[26] |
Y. Han, G. Lu and E. Sawyer,
Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group, Anal. PDE, 7 (2014), 1465-1534.
doi: 10.2140/apde.2014.7.1465. |
[27] |
Q. Hong and G. Lu,
Weighted $ L^p $ estimates for rough bi-parameter Fourier integral operators, J. Differential Equations, 265 (2018), 1097-1127.
doi: 10.1016/j.jde.2018.03.024. |
[28] |
Q. Hong, G. Lu and L. Zhang,
$ L^p $ boundedness of rough bi-parameter Fourier integral operators, Forum Math., 30 (2018), 87-107.
doi: 10.1515/forum-2016-0221. |
[29] |
Q. Hong and L. Zhang,
$ L^p $ estimates for bi-parameter and bilinear Fourier integral operators, Acta Math. Sin. (Engl. Ser.), 33 (2017), 165-186.
doi: 10.1007/s10114-016-6269-6. |
[30] |
B. Jessen, J. Marcinkiewicz and A. Zygmund,
Note on the differentiability of multiple integrals, Fundamenta Mathematicae, 25 (1935), 217-234.
|
[31] |
J. L. Journé,
Calderón-Zygmund operators on product spaces, Rev. Mat. Iberoamericana, 1 (1985), 55-91.
doi: 10.4171/RMI/15. |
[32] |
J. L. Journé,
Two problems of Calderón-Zygmund theory on product spaces, Ann. Inst. Fourier (Grenoble), 38 (1988), 111-132.
doi: 10.5802/aif.1125. |
[33] |
M. Kobayashi and Y. Sawano,
Molecular decomposition of the modulation spaces, Osaka J. Math., 47 (2010), 1029-1053.
doi: 10.1007/s11072-010-0114-0. |
[34] |
M. Kobayashi, M. Sugimoto and N. Tomita,
Trace ideals for pseudo-differential operators and their commutators with symbols in $ \alpha $-modulation spaces, J. Anal. Math., 107 (2009), 141-160.
doi: 10.1007/s11854-009-0006-3. |
[35] |
M. Kobayashi and M. Sugimoto,
The inclusion relation between Sobolev and modulation spaces, J. Funct. Anal., 260 (2011), 3189-3208.
doi: 10.1016/j.jfa.2011.02.015. |
[36] |
G. Lu and Z. Ruan,
Duality theory of weighted Hardy spaces with arbitrary number of parameters, Forum Math., 26 (2014), 1429-1457.
doi: 10.1515/forum-2012-0018. |
[37] |
G. Lu and Y. Zhu,
Singular integrals and weighted Triebel-Lizorkin and Besov spaces of arbitrary number of parameters, Acta Math. Sin. (Engl. Ser.), 29 (2013), 39-52.
doi: 10.1007/s10114-012-1402-7. |
[38] |
C. Muscalu, J. Pipher, T. Tao and C. Thiele,
Bi-parameter paraproducts, Acta Math., 193 (2004), 269-296.
doi: 10.1007/BF02392566. |
[39] |
C. Muscalu, J. Pipher, T. Tao and C. Thiele,
Multi-parameter paraproducts, Rev. Mat. Iberoam., 22 (2006), 963-976.
doi: 10.4171/RMI/480. |
[40] |
D. Müller, F. Ricci and E. M. Stein,
Marcinkiewicz multiplers and multiparameter structure on Heisenberg (-type) groups, Ⅰ, Invent. Math., 119 (1995), 119-233.
doi: 10.1007/BF01245180. |
[41] |
D. Müller, F. Ricci and and E. M. Stein,
Marcinkiewicz multipliers and multi-parameter structure on Heisenberg(-type) groups, Ⅱ, Math. Z., 221 (1996), 267-291.
doi: 10.1007/PL00022737. |
[42] |
A. Nagel, F. Ricci and E. M. Stein,
Singular integrals with flag kernels and analysis on quadratic CR manifolds, J. Funct. Anal., 181 (2001), 29-118.
doi: 10.1006/jfan.2000.3714. |
[43] |
A. Nagel, F. Ricci, E. M. Stein and S. Wainger,
Singular integrals with flag kernels on homogeneous groups, Ⅰ, Rev. Mat. Iberoam., 28 (2012), 631-722.
doi: 10.4171/rmi/688. |
[44] |
A. Nagel, F. Ricci, E. M. Stein and S. Wainger, Algebras of singular integral operators with kernels controlled by multiple norms, Mem. Amer. Math. Soc., 256 (2018), no. 1230, vii+141 pp.
doi: 10.1090/memo/1230. |
[45] |
J. Pipher,
Journé's covering lemma and its extension to higher dimensions, Duke Math. J., 53 (1986), 683-690.
doi: 10.1215/S0012-7094-86-05337-8. |
[46] |
Z. Ruan,
The Calderón-Zygmund decomposition and interpolation on weighted Hardy spaces, Acta Math. Sin. (Engl. Ser.), 27 (2011), 1967-1978.
doi: 10.1007/s10114-011-9338-x. |
[47] |
J. Sjöstrand,
An algebra of pseudodifferential operators, Math. Res. Lett., 1 (1994), 185-192.
doi: 10.4310/MRL.1994.v1.n2.a6. |
[48] |
B. Street, Multi-parameter Singular Integrals, Annals of Mathematics Studies, 189, Princeton University Press, Princeton, NJ, 2014.
doi: 10.1515/9781400852758.![]() ![]() ![]() |
[49] |
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. |
[50] |
K. Tachizawa,
The boundedness of pseudodifferential operators on modulation spaces, Math. Nachr., 168 (1994), 263-277.
doi: 10.1002/mana.19941680116. |
[51] |
N. Tomita,
On the Hörmander multiplier theorem and modulation spaces, Appl. Comput. Harmon. Anal., 26 (2009), 408-415.
doi: 10.1016/j.acha.2008.10.001. |
[52] |
H. Triebel, Theory of Function Spaces, Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983.
doi: 10.1007/978-3-0346-0416-1. |
[53] |
B. Wang and H. Hudzik,
The global Cauchy problem for the NLS and NLKG with small rough data, J. Differential Equations, 231 (2007), 36-73.
doi: 10.1016/j.jde.2006.09.004. |
[54] |
C. Xu and L. Huang,
Boundedness of bi-parameter pseudo-differential operators on bi-parameter $ \alpha $-modulation spaces, Nonlinear Anal., 180 (2019), 20-40.
doi: 10.1016/j.na.2018.09.004. |
[55] |
C. Xu, Boundedness of bi-parameter fractional integrals on bi-parameter modulation spaces, preprint. |
show all references
References:
[1] |
R. Balan, P. G. Casazza, C. Heil and Z. Landau,
Density, overcompleteness, and localization of frames, Ⅱ, Gabor systems, J. Fourier Anal. Appl., 12 (2006), 309-344.
doi: 10.1007/s00041-005-5035-4. |
[2] |
Á. Bényi, L. Grafakos, K. Gröchenig and K. Okoudjou,
A class of Fourier multipliers for modulation spaces, Appl. Comput. Harmon. Anal., 19 (2005), 131-139.
doi: 10.1016/j.acha.2005.02.002. |
[3] |
Á. Bényi and K. Okoudjou,
Local well-posedness of nonlinear dispersive equations on modulation spaces, Bull. Lond. Math. Soc., 41 (2009), 549-558.
doi: 10.1112/blms/bdp027. |
[4] |
S-Y. A. Chang and R. Fefferman,
Some recent developments in Fourier analysis and $ H^{p} $ theory on product domains, Bull. Amer. Math. Soc., 12 (1985), 1-43.
doi: 10.1090/S0273-0979-1985-15291-7. |
[5] |
S-Y. A. Chang and R. Fefferman,
The Calderón-Zygmund decomposition on product domains, Amer. J. Math., 104 (1982), 455-468.
doi: 10.2307/2374150. |
[6] |
S-Y. A. Chang and R. Fefferman,
A continuous version of duality of $H^{1}$ with $BMO$ on the bidisc, Ann. of Math., 112 (1980), 179-201.
doi: 10.2307/1971324. |
[7] |
J. Chen,
Hörmander type theorem for Fourier multipliers with optimal smoothness on Hardy spaces of arbitrary number of parameters, Acta Math. Sin. (Engl. Ser.), 33 (2017), 1083-1106.
doi: 10.1007/s10114-017-6526-3. |
[8] |
J. Chen and G. Lu,
Hörmander type theorems for multi-linear and multi-parameter Fourier multiplier operators with limited smoothness, Nonlinear Anal., 101 (2014), 98-112.
doi: 10.1016/j.na.2014.01.005. |
[9] |
J. Chen and G. Lu,
Hörmander type theorem on bi-parameter Hardy spaces for bi-parameter Fourier multipliers with optimal smoothness, Rev. Mat. Iberoam., 34 (2018), 1541-1561.
doi: 10.4171/rmi/1035. |
[10] |
W. Ding and G. Lu,
Duality of multi-parameter Triebel-Lizorkin spaces associated with the composition of two singular integral operators, Tans. Amer. Math. Soc., 368 (2016), 7119-7152.
doi: 10.1090/tran/6576. |
[11] |
Y. Ding, G. Lu and B. Ma,
Multi-parameter Triebel-Lizorkin and Besov spaces associated with flag singular integrals, Acta Math. Sin. (Engl. Ser.), 26 (2010), 603-620.
doi: 10.1007/s10114-010-8352-8. |
[12] |
R. Fefferman,
Harmonic Analysis on product spaces, Ann. of Math., 126 (1987), 109-130.
doi: 10.2307/1971346. |
[13] |
R. Fefferman and J. Pipher,
Multiparameter operators and sharp weighted inequalities, Amer. J. Math., 119 (1997), 337-369.
doi: 10.1353/ajm.1997.0011. |
[14] |
R. Fefferman and E. M. Stein,
Singular integrals on product spaces, Adv. Math., 45 (1982), 117-143.
doi: 10.1016/S0001-8708(82)80001-7. |
[15] |
H. G. Feichtinger, Modulation spaces on locally compact abelian groups, Technical report, University Vienna, January 1983. |
[16] |
G. Feichtinger and K. Gröchenig,
Gabor frames and time-frequency analysis of distributions, J. Funct. Anal., 146 (1997), 464-495.
doi: 10.1006/jfan.1996.3078. |
[17] |
H. G. Feichtinger and G. Narimani,
Fourier multipliers of classical modulation spaces, Appl. Comput. Harmon. Anal., 21 (2006), 349-359.
doi: 10.1016/j.acha.2006.04.010. |
[18] |
S. H. Ferguson and M. T. Lacey,
A characterization of product BMO by commutators, Acta Math., 189 (2002), 143-160.
doi: 10.1007/BF02392840. |
[19] |
K. Gröchenig, Foundations of Time-frequency Analysis, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, 2001.
doi: 10.1007/BF02392840. |
[20] |
K. Gröchenig and C. Heil,
Modulation spaces and pseudodifferential operators, Integral Equations Operator Theory, 34 (1999), 439-457.
doi: 10.1007/BF01272884. |
[21] |
K. Gröchenig and Z. Rzeszotnik,
Banach algebras of pseudodifferential operators and their almost diagonalization, Ann. Inst. Fourier (Grenoble), 58 (2008), 2279-2314.
doi: 10.5802/aif.2414. |
[22] |
R. Gundy and E. M. Stein,
$ H^{p} $ theory for the polydisk, Proc. Nat. Acad. Sci., 76 (1979), 1026-1029.
doi: 10.1073/pnas.76.3.1026. |
[23] |
Y. Han, J. Li and G. Lu,
Multiparameter Hardy space theory on Carnot-Carathéodory spaces and product spaces of homogeneous type, Trans. Amer. Math. Soc., 365 (2013), 319-360.
doi: 10.1090/S0002-9947-2012-05638-8. |
[24] |
Y. Han and G. Lu, Discrete Littlewood-Paley-Stein theory and multi-parameter Hardy spaces associated with the flag singular integrals, preprint, arXiv: 0801.1701.
doi: 10.1090/S0002-9947-2012-05638-8. |
[25] |
Y. Han, G. Lu and Z. Ruan,
Boundedness criterion of Journé's class of singular integrals on multiparameter Hardy spaces, J. Funct. Anal., 264 (2013), 1238-1268.
doi: 10.1016/j.jfa.2012.12.006. |
[26] |
Y. Han, G. Lu and E. Sawyer,
Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group, Anal. PDE, 7 (2014), 1465-1534.
doi: 10.2140/apde.2014.7.1465. |
[27] |
Q. Hong and G. Lu,
Weighted $ L^p $ estimates for rough bi-parameter Fourier integral operators, J. Differential Equations, 265 (2018), 1097-1127.
doi: 10.1016/j.jde.2018.03.024. |
[28] |
Q. Hong, G. Lu and L. Zhang,
$ L^p $ boundedness of rough bi-parameter Fourier integral operators, Forum Math., 30 (2018), 87-107.
doi: 10.1515/forum-2016-0221. |
[29] |
Q. Hong and L. Zhang,
$ L^p $ estimates for bi-parameter and bilinear Fourier integral operators, Acta Math. Sin. (Engl. Ser.), 33 (2017), 165-186.
doi: 10.1007/s10114-016-6269-6. |
[30] |
B. Jessen, J. Marcinkiewicz and A. Zygmund,
Note on the differentiability of multiple integrals, Fundamenta Mathematicae, 25 (1935), 217-234.
|
[31] |
J. L. Journé,
Calderón-Zygmund operators on product spaces, Rev. Mat. Iberoamericana, 1 (1985), 55-91.
doi: 10.4171/RMI/15. |
[32] |
J. L. Journé,
Two problems of Calderón-Zygmund theory on product spaces, Ann. Inst. Fourier (Grenoble), 38 (1988), 111-132.
doi: 10.5802/aif.1125. |
[33] |
M. Kobayashi and Y. Sawano,
Molecular decomposition of the modulation spaces, Osaka J. Math., 47 (2010), 1029-1053.
doi: 10.1007/s11072-010-0114-0. |
[34] |
M. Kobayashi, M. Sugimoto and N. Tomita,
Trace ideals for pseudo-differential operators and their commutators with symbols in $ \alpha $-modulation spaces, J. Anal. Math., 107 (2009), 141-160.
doi: 10.1007/s11854-009-0006-3. |
[35] |
M. Kobayashi and M. Sugimoto,
The inclusion relation between Sobolev and modulation spaces, J. Funct. Anal., 260 (2011), 3189-3208.
doi: 10.1016/j.jfa.2011.02.015. |
[36] |
G. Lu and Z. Ruan,
Duality theory of weighted Hardy spaces with arbitrary number of parameters, Forum Math., 26 (2014), 1429-1457.
doi: 10.1515/forum-2012-0018. |
[37] |
G. Lu and Y. Zhu,
Singular integrals and weighted Triebel-Lizorkin and Besov spaces of arbitrary number of parameters, Acta Math. Sin. (Engl. Ser.), 29 (2013), 39-52.
doi: 10.1007/s10114-012-1402-7. |
[38] |
C. Muscalu, J. Pipher, T. Tao and C. Thiele,
Bi-parameter paraproducts, Acta Math., 193 (2004), 269-296.
doi: 10.1007/BF02392566. |
[39] |
C. Muscalu, J. Pipher, T. Tao and C. Thiele,
Multi-parameter paraproducts, Rev. Mat. Iberoam., 22 (2006), 963-976.
doi: 10.4171/RMI/480. |
[40] |
D. Müller, F. Ricci and E. M. Stein,
Marcinkiewicz multiplers and multiparameter structure on Heisenberg (-type) groups, Ⅰ, Invent. Math., 119 (1995), 119-233.
doi: 10.1007/BF01245180. |
[41] |
D. Müller, F. Ricci and and E. M. Stein,
Marcinkiewicz multipliers and multi-parameter structure on Heisenberg(-type) groups, Ⅱ, Math. Z., 221 (1996), 267-291.
doi: 10.1007/PL00022737. |
[42] |
A. Nagel, F. Ricci and E. M. Stein,
Singular integrals with flag kernels and analysis on quadratic CR manifolds, J. Funct. Anal., 181 (2001), 29-118.
doi: 10.1006/jfan.2000.3714. |
[43] |
A. Nagel, F. Ricci, E. M. Stein and S. Wainger,
Singular integrals with flag kernels on homogeneous groups, Ⅰ, Rev. Mat. Iberoam., 28 (2012), 631-722.
doi: 10.4171/rmi/688. |
[44] |
A. Nagel, F. Ricci, E. M. Stein and S. Wainger, Algebras of singular integral operators with kernels controlled by multiple norms, Mem. Amer. Math. Soc., 256 (2018), no. 1230, vii+141 pp.
doi: 10.1090/memo/1230. |
[45] |
J. Pipher,
Journé's covering lemma and its extension to higher dimensions, Duke Math. J., 53 (1986), 683-690.
doi: 10.1215/S0012-7094-86-05337-8. |
[46] |
Z. Ruan,
The Calderón-Zygmund decomposition and interpolation on weighted Hardy spaces, Acta Math. Sin. (Engl. Ser.), 27 (2011), 1967-1978.
doi: 10.1007/s10114-011-9338-x. |
[47] |
J. Sjöstrand,
An algebra of pseudodifferential operators, Math. Res. Lett., 1 (1994), 185-192.
doi: 10.4310/MRL.1994.v1.n2.a6. |
[48] |
B. Street, Multi-parameter Singular Integrals, Annals of Mathematics Studies, 189, Princeton University Press, Princeton, NJ, 2014.
doi: 10.1515/9781400852758.![]() ![]() ![]() |
[49] |
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. |
[50] |
K. Tachizawa,
The boundedness of pseudodifferential operators on modulation spaces, Math. Nachr., 168 (1994), 263-277.
doi: 10.1002/mana.19941680116. |
[51] |
N. Tomita,
On the Hörmander multiplier theorem and modulation spaces, Appl. Comput. Harmon. Anal., 26 (2009), 408-415.
doi: 10.1016/j.acha.2008.10.001. |
[52] |
H. Triebel, Theory of Function Spaces, Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983.
doi: 10.1007/978-3-0346-0416-1. |
[53] |
B. Wang and H. Hudzik,
The global Cauchy problem for the NLS and NLKG with small rough data, J. Differential Equations, 231 (2007), 36-73.
doi: 10.1016/j.jde.2006.09.004. |
[54] |
C. Xu and L. Huang,
Boundedness of bi-parameter pseudo-differential operators on bi-parameter $ \alpha $-modulation spaces, Nonlinear Anal., 180 (2019), 20-40.
doi: 10.1016/j.na.2018.09.004. |
[55] |
C. Xu, Boundedness of bi-parameter fractional integrals on bi-parameter modulation spaces, preprint. |
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