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November  2019, 18(6): 3103-3120. doi: 10.3934/cpaa.2019139

Molecular decomposition and a class of Fourier multipliers for bi-parameter modulation spaces

Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China

* Corresponding author

Received  October 2018 Revised  February 2019 Published  May 2019

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 [33]. Furthermore, we prove the boundedness of a class of Fourier multipliers on bi-parameter modulation spaces, generalizing the results of Bényi et al. [2] and Feichtinger and Narimani [17].

Citation: Qing Hong, Guorong Hu. Molecular decomposition and a class of Fourier multipliers for bi-parameter modulation spaces. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3103-3120. doi: 10.3934/cpaa.2019139
References:
[1]

R. BalanP. G. CasazzaC. 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. BényiL. GrafakosK. 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. DingG. 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. HanJ. 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. HanG. 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. HanG. 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. HongG. 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. JessenJ. 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. KobayashiM. 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. MuscaluJ. PipherT. Tao and C. Thiele, Bi-parameter paraproducts, Acta Math., 193 (2004), 269-296. doi: 10.1007/BF02392566.

[39]

C. MuscaluJ. PipherT. Tao and C. Thiele, Multi-parameter paraproducts, Rev. Mat. Iberoam., 22 (2006), 963-976. doi: 10.4171/RMI/480.

[40]

D. MüllerF. 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. MüllerF. 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. NagelF. 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. NagelF. RicciE. 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. BalanP. G. CasazzaC. 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. BényiL. GrafakosK. 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. DingG. 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. HanJ. 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. HanG. 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. HanG. 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. HongG. 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. JessenJ. 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. KobayashiM. 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. MuscaluJ. PipherT. Tao and C. Thiele, Bi-parameter paraproducts, Acta Math., 193 (2004), 269-296. doi: 10.1007/BF02392566.

[39]

C. MuscaluJ. PipherT. Tao and C. Thiele, Multi-parameter paraproducts, Rev. Mat. Iberoam., 22 (2006), 963-976. doi: 10.4171/RMI/480.

[40]

D. MüllerF. 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. MüllerF. 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. NagelF. 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. NagelF. RicciE. 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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