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Random attractor for the 2D stochastic nematic liquid crystals flows
Molecular characterization of anisotropic weak Musielak-Orlicz Hardy spaces and their applications
College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China |
Let $ A $ be a real $ n\times n $ matrix with all its eigenvalues $ \lambda $ satisfy $ |\lambda|>1 $. Let $ \varphi: \mathbb{R}^n\times[0, \, \infty)\to[0, \, \infty) $ be an anisotropic Musielak-Orlicz function, i.e., $ \varphi(x, \cdot) $ is an Orlicz function uniformly in $ x\in{\mathbb{R}^n} $ and $ \varphi(\cdot, \, t) $ is an anisotropic Muckenhoupt $ \mathcal {A}_\infty({\mathbb{R}^n}) $ weight uniformly in $ t\in(0, \, \infty) $. In this article, the authors introduce the anisotropic weak Musielak-Orlicz Hardy space $ WH^{\varphi}_A(\mathbb{R}^n) $ via the grand maximal function and establish its molecular characterization which are anisotropic extensions of Liang, Yang and Jiang (Math. Nachr. 289: 634-677, 2016). As an application, the boundedness of anisotropic Calderón-Zygmund operators from $ H_A^\varphi(\mathbb{R}^n) $ to $ WH_A^\varphi(\mathbb{R}^n) $ in the critical case is presented.
References:
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J. Álvarez, Hp and weak Hp continuity of Calderón-Zygmund type operators, in Fourier Analysis (Orono, ME, 1992), Lecture Notes in Pure and Appl. Math., 157 (1994), Dekker, New York, 17–34. |
[2] |
J. Álvarez,
Continuity properties for linear commutators of Calderón-Zygmund operators, Collect. Math., 49 (1998), 17-31.
|
[3] |
Z. Birnbaum and W. Orlicz,
Über die verallgemeinerung des begriffes der zueinander konjugierten potenzen (German), Studia Math., 3 (1931), 1-67.
|
[4] |
A. Bonami, J. Feuto and S. Grellier,
Endpoint for the DIV-CURL lemma in Hardy spaces, Publ. Mat., 54 (2010), 341-358.
doi: 10.5565/PUBLMAT_54210_03. |
[5] |
M. Bownik, Anisotropic Hardy spaces and wavelets, Mem. Amer. Math. Soc., 164 (2003), ⅵ+122 pp.
doi: 10.1090/memo/0781. |
[6] |
M. Bownik and K.-P. Ho,
Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces, Tran. Amer. Math. Soc., 358 (2006), 1469-1510.
doi: 10.1090/S0002-9947-05-03660-3. |
[7] |
M. Bownik, B. Li, D. Yang and Y. Zhou,
Weighted anisotropic Hardy spaces and their applications in boundedness of sublinear operators, Indiana Univ. Math. J., 57 (2008), 3065-3100.
doi: 10.1512/iumj.2008.57.3414. |
[8] |
M. Bownik, B. Li, D. Yang and Y. Zhou,
Weighted anisotropic product Hardy spaces and boundedness of sublinear operators, Math. Nachr., 283 (2010), 392-442.
doi: 10.1002/mana.200910078. |
[9] |
T. Bui, J. Cao, L. D. Ky, D. Yang and S. Yang,
Musielak-Orlicz Hardy spaces associated with operators satisfying reinforced off-diagonal estimates, Anal. Geom. Metr. Spaces, 1 (2013), 69-129.
doi: 10.2478/agms-2012-0006. |
[10] |
A. P. Calderón and A. Torchinsky,
Parabolic maximal function associated with a distribution, Adv. Math., 16 (1975), 1-64.
doi: 10.1016/0001-8708(75)90099-7. |
[11] |
J. Cao, D.-C. Chang, D. Yang and S. Yang,
Boundedness of second order Riesz transforms associated to Schrödinger operators on Musielak-Orlicz-Hardy spaces, Commun. Pure Appl. Anal., 13 (2014), 1435-1463.
doi: 10.3934/cpaa.2014.13.1435. |
[12] |
W. Chen and Y. Lai,
Boundedness of fractional integrals in Hardy spaces with non-doubling measure, Anal. Theory Appl., 22 (2006), 195-200.
doi: 10.1007/BF03218712. |
[13] |
R. R. Coifman and G. Weiss, Analyse Harmonique Non-commutative sur Certains Espaces Homogènes (French) [Non-commutative harmonic analysis on certain homogeneous spaces] Lecture Notes in Math., 242, Springer, Berlin, 1971. |
[14] |
L. Diening, Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math., 129 (2005), 657–700.
doi: 10.1016/j.bulsci.2003.10.003. |
[15] |
Y. Ding and S. Lan,
Anisotropic weak Hardy spaces and interpolation theorems, Sci. China Math., 51 (2008), 1690-1704.
doi: 10.1007/s11425-008-0009-z. |
[16] |
Y. Ding and S. Lan,
Anisotropic Hardy estimates for multilinear operators, Adv. Math., 38 (2009), 168-184.
|
[17] |
Y. Ding and S. Lu, Hardy spaces estimates for multilinear operators with homogeneous kernels, Nagoya Math. J., 170 (2003), 117–133.
doi: 10.1017/S0027763000008552. |
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Y. Ding, S. Lu and S. Shao,
Integral operators with variable kernels on weak Hardy spaces, J. Math. Anal. Appl., 317 (2006), 127-135.
doi: 10.1016/j.jmaa.2005.10.085. |
[19] |
R. Fefferman and F. Soria,
The spaces weak H1, Studia Math., 85 (1986), 1-16.
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J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam, 1985. |
[21] |
L. Grafakos,
Hardy space estimates for multilinear operators, Ⅱ, Rev. Mat. Iberoam., 8 (1992), 69-92.
doi: 10.4171/RMI/117. |
[22] |
L. Grafakos, Modern Fourier Analysis, 2ed edition, Graduate Texts in Mathematics 250, Springer, New York, 2009.
doi: 10.1007/978-0-387-09434-2. |
[23] |
S. Hou, D. Yang and S. Yang, Lusin area function and molecular characterizations of MusielakOrlicz Hardy spaces and their applications, Commun. Contemp. Math., 15 (2013), 1350029, 37 pp.
doi: 10.1142/S0219199713500296. |
[24] |
R. Johnson and C. J. Neugebauer,
Homeomorphisms preserving Ap, Rev. Mat. Iberoam., 3 (1987), 249-273.
doi: 10.4171/RMI/50. |
[25] |
N. J. Kalton,
Linear operators on Lp for 0 < p < 1, Trans. Amer. Math. Soc., 259 (1980), 319-355.
doi: 10.2307/1998234. |
[26] |
L. D. Ky,
Bilinear decompositions and commutators of singular integral operators, Trans. Amer. Math. Soc., 365 (2013), 2931-2958.
doi: 10.1090/S0002-9947-2012-05727-8. |
[27] |
L. D. Ky,
New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators, Integral Equations Operator Theory, 78 (2014), 115-150.
doi: 10.1007/s00020-013-2111-z. |
[28] |
L. D. Ky, Bilinear decompositions for the product space $H^1_L\times BMO_L$, Math. Nachr., 287 (2014), 1288–1297.
doi: 10.1002/mana.201200101. |
[29] |
B. Li, X. Fan and D. Yang,
Littlewood-Paley theory of anisotropic Hardy spaces of Musielak-Orlicz type, Taiwanese J. Math., 19 (2015), 279-314.
doi: 10.11650/tjm.19.2015.4692. |
[30] |
B. Li, X. Fan, Z. Fu and D. Yang,
Molecular characterization of anisotropic Musielak-Orlicz Hardy spaces and their applications, Acta Math. Sin., 32 (2016), 1391-1414.
doi: 10.1007/s10114-016-4741-y. |
[31] |
B. Li, R. Sun, M. Liao and B. Li, Littlewood-Paley characterizations of anisotropic weak Musielak-Orlicz Hardy spaces, Nagoya Math. J., 1–40.
doi: 10.1017/nmj.2018.10. |
[32] |
B. Li, D. Yang and W. Yuan, Anisotropic Hardy spaces of Musielak-Orlicz type with applications to boundedness of sublinear operators, The Scientific World Journal, (2014), 19 pp. |
[33] |
Y. Liang, J. Huang and D. Yang,
New real-variable characterizations of Musielak-Orlicz Hardy spaces, J. Math. Anal. Appl., 395 (2012), 413-428.
doi: 10.1016/j.jmaa.2012.05.049. |
[34] |
Y. Liang, D. Yang and R. Jiang,
Weak Musielak-Orlicz Hardy spaces and applications, Math. Nachr., 289 (2016), 637-677.
doi: 10.1002/mana.201500152. |
[35] |
H. Liu, The weak Hp spaces on homogeneous groups, in Harminic Analysis (Tianjin, 1998), Lecture Notes in Math. Vol., 1984 (Springer, Berlin, 1991), pp. 113–118.
doi: 10.1007/BFb0087762. |
[36] |
J. Liu, F. Weisz, D. Yang and W. Yuan, Littlewood-Paley and finite atomic characterizations of anisotropic variable Hardy-Lorentz spaces and their applications, J. Fourier Anal. Appl., (2018).
doi: 10.1007/s00041-018-9609-3. |
[37] |
J. Liu, D. Yang and W. Yuan,
Anisotropic Hardy-Lorentz spaces and their applications, Sci. China Math., 59 (2016), 1669-1720.
doi: 10.1007/s11425-016-5157-y. |
[38] |
J. Liu, D. Yang and W. Yuan,
Anisotropic variable Hardy-Lorentz spaces and their real interpolation, J. Math. Anal. Appl., 456 (2017), 356-393.
doi: 10.1016/j.jmaa.2017.07.003. |
[39] |
J. Musielak, Orlicz Spaces and Modular Spaces, Springer-Verlag, Berlin-NewYork, 1983.
doi: 10.1007/BFb0072210. |
[40] |
W. Orlicz,
Über eine gewisse Klasse von Raumen vom Typus B (German), Bull. Int. Acad. Pol. Ser. A, 8 (1932), 207-220.
|
[41] |
T. Quek and D. Yang,
Calderón-Zygmund-type operators on weighted weak Hardy spaces over ${{\mathbb R}^n}$, Acta Math. Sin., 16 (2000), 141-160.
doi: 10.1007/s101149900022. |
[42] |
E. M. Stein, M. Taibleson and G. Weiss,
Weak type estimates for maximal operators on certain Hp classes, Rend. Circ. Mat. Palermo, 1 (1981), 81-97.
|
[43] |
J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Berlin-New York: Springer-Verlag, 1989.
doi: 10.1007/BFb0091154. |
[44] |
M. H. Taibleson and G. Weiss, The molecular characterization of certain Hardy spaces, Asterisque, Mathematical Reviews, 77 (1980), 67–149. |
[45] |
L. Tang,
The weighted weak local Hardy spaces, Rocky Mountain J. Math., 44 (2014), 297-315.
doi: 10.1216/RMJ-2014-44-1-297. |
[46] |
H. Wang, Boundedness of several integral operators with bounded variable kernels on Hardy and weak Hardy spaces, Internat. J. Math., 24 (2013), 1350095, 22 pp.
doi: 10.1142/S0129167X1350095X. |
[47] |
D. Yang and S. Yang,
Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated to non-negative self-adjoint operators satisfying Gaussian estimates, Commun. Pure Appl. Anal., 15 (2016), 2135-2160.
doi: 10.3934/cpaa.2016031. |
[48] |
D. Yang, Y. Liang and L. D. Ky, Real-Variable Theory of Musielak-Orlicz Hardy Spaces, Lecture Notes in Mathematics, 2182, Springer-Verlag, Cham, 2017.
doi: 10.1007/978-3-319-54361-1. |
[49] |
H. Zhang, C. Qi and B. Li,
Anisotropic weak Hardy spaces of Musielak-Orlicz type and their applications, Front. Math. China, 12 (2017), 993-1022.
doi: 10.1007/s11464-016-0546-7. |
show all references
References:
[1] |
J. Álvarez, Hp and weak Hp continuity of Calderón-Zygmund type operators, in Fourier Analysis (Orono, ME, 1992), Lecture Notes in Pure and Appl. Math., 157 (1994), Dekker, New York, 17–34. |
[2] |
J. Álvarez,
Continuity properties for linear commutators of Calderón-Zygmund operators, Collect. Math., 49 (1998), 17-31.
|
[3] |
Z. Birnbaum and W. Orlicz,
Über die verallgemeinerung des begriffes der zueinander konjugierten potenzen (German), Studia Math., 3 (1931), 1-67.
|
[4] |
A. Bonami, J. Feuto and S. Grellier,
Endpoint for the DIV-CURL lemma in Hardy spaces, Publ. Mat., 54 (2010), 341-358.
doi: 10.5565/PUBLMAT_54210_03. |
[5] |
M. Bownik, Anisotropic Hardy spaces and wavelets, Mem. Amer. Math. Soc., 164 (2003), ⅵ+122 pp.
doi: 10.1090/memo/0781. |
[6] |
M. Bownik and K.-P. Ho,
Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces, Tran. Amer. Math. Soc., 358 (2006), 1469-1510.
doi: 10.1090/S0002-9947-05-03660-3. |
[7] |
M. Bownik, B. Li, D. Yang and Y. Zhou,
Weighted anisotropic Hardy spaces and their applications in boundedness of sublinear operators, Indiana Univ. Math. J., 57 (2008), 3065-3100.
doi: 10.1512/iumj.2008.57.3414. |
[8] |
M. Bownik, B. Li, D. Yang and Y. Zhou,
Weighted anisotropic product Hardy spaces and boundedness of sublinear operators, Math. Nachr., 283 (2010), 392-442.
doi: 10.1002/mana.200910078. |
[9] |
T. Bui, J. Cao, L. D. Ky, D. Yang and S. Yang,
Musielak-Orlicz Hardy spaces associated with operators satisfying reinforced off-diagonal estimates, Anal. Geom. Metr. Spaces, 1 (2013), 69-129.
doi: 10.2478/agms-2012-0006. |
[10] |
A. P. Calderón and A. Torchinsky,
Parabolic maximal function associated with a distribution, Adv. Math., 16 (1975), 1-64.
doi: 10.1016/0001-8708(75)90099-7. |
[11] |
J. Cao, D.-C. Chang, D. Yang and S. Yang,
Boundedness of second order Riesz transforms associated to Schrödinger operators on Musielak-Orlicz-Hardy spaces, Commun. Pure Appl. Anal., 13 (2014), 1435-1463.
doi: 10.3934/cpaa.2014.13.1435. |
[12] |
W. Chen and Y. Lai,
Boundedness of fractional integrals in Hardy spaces with non-doubling measure, Anal. Theory Appl., 22 (2006), 195-200.
doi: 10.1007/BF03218712. |
[13] |
R. R. Coifman and G. Weiss, Analyse Harmonique Non-commutative sur Certains Espaces Homogènes (French) [Non-commutative harmonic analysis on certain homogeneous spaces] Lecture Notes in Math., 242, Springer, Berlin, 1971. |
[14] |
L. Diening, Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math., 129 (2005), 657–700.
doi: 10.1016/j.bulsci.2003.10.003. |
[15] |
Y. Ding and S. Lan,
Anisotropic weak Hardy spaces and interpolation theorems, Sci. China Math., 51 (2008), 1690-1704.
doi: 10.1007/s11425-008-0009-z. |
[16] |
Y. Ding and S. Lan,
Anisotropic Hardy estimates for multilinear operators, Adv. Math., 38 (2009), 168-184.
|
[17] |
Y. Ding and S. Lu, Hardy spaces estimates for multilinear operators with homogeneous kernels, Nagoya Math. J., 170 (2003), 117–133.
doi: 10.1017/S0027763000008552. |
[18] |
Y. Ding, S. Lu and S. Shao,
Integral operators with variable kernels on weak Hardy spaces, J. Math. Anal. Appl., 317 (2006), 127-135.
doi: 10.1016/j.jmaa.2005.10.085. |
[19] |
R. Fefferman and F. Soria,
The spaces weak H1, Studia Math., 85 (1986), 1-16.
doi: 10.4064/sm-85-1-1-16. |
[20] |
J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam, 1985. |
[21] |
L. Grafakos,
Hardy space estimates for multilinear operators, Ⅱ, Rev. Mat. Iberoam., 8 (1992), 69-92.
doi: 10.4171/RMI/117. |
[22] |
L. Grafakos, Modern Fourier Analysis, 2ed edition, Graduate Texts in Mathematics 250, Springer, New York, 2009.
doi: 10.1007/978-0-387-09434-2. |
[23] |
S. Hou, D. Yang and S. Yang, Lusin area function and molecular characterizations of MusielakOrlicz Hardy spaces and their applications, Commun. Contemp. Math., 15 (2013), 1350029, 37 pp.
doi: 10.1142/S0219199713500296. |
[24] |
R. Johnson and C. J. Neugebauer,
Homeomorphisms preserving Ap, Rev. Mat. Iberoam., 3 (1987), 249-273.
doi: 10.4171/RMI/50. |
[25] |
N. J. Kalton,
Linear operators on Lp for 0 < p < 1, Trans. Amer. Math. Soc., 259 (1980), 319-355.
doi: 10.2307/1998234. |
[26] |
L. D. Ky,
Bilinear decompositions and commutators of singular integral operators, Trans. Amer. Math. Soc., 365 (2013), 2931-2958.
doi: 10.1090/S0002-9947-2012-05727-8. |
[27] |
L. D. Ky,
New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators, Integral Equations Operator Theory, 78 (2014), 115-150.
doi: 10.1007/s00020-013-2111-z. |
[28] |
L. D. Ky, Bilinear decompositions for the product space $H^1_L\times BMO_L$, Math. Nachr., 287 (2014), 1288–1297.
doi: 10.1002/mana.201200101. |
[29] |
B. Li, X. Fan and D. Yang,
Littlewood-Paley theory of anisotropic Hardy spaces of Musielak-Orlicz type, Taiwanese J. Math., 19 (2015), 279-314.
doi: 10.11650/tjm.19.2015.4692. |
[30] |
B. Li, X. Fan, Z. Fu and D. Yang,
Molecular characterization of anisotropic Musielak-Orlicz Hardy spaces and their applications, Acta Math. Sin., 32 (2016), 1391-1414.
doi: 10.1007/s10114-016-4741-y. |
[31] |
B. Li, R. Sun, M. Liao and B. Li, Littlewood-Paley characterizations of anisotropic weak Musielak-Orlicz Hardy spaces, Nagoya Math. J., 1–40.
doi: 10.1017/nmj.2018.10. |
[32] |
B. Li, D. Yang and W. Yuan, Anisotropic Hardy spaces of Musielak-Orlicz type with applications to boundedness of sublinear operators, The Scientific World Journal, (2014), 19 pp. |
[33] |
Y. Liang, J. Huang and D. Yang,
New real-variable characterizations of Musielak-Orlicz Hardy spaces, J. Math. Anal. Appl., 395 (2012), 413-428.
doi: 10.1016/j.jmaa.2012.05.049. |
[34] |
Y. Liang, D. Yang and R. Jiang,
Weak Musielak-Orlicz Hardy spaces and applications, Math. Nachr., 289 (2016), 637-677.
doi: 10.1002/mana.201500152. |
[35] |
H. Liu, The weak Hp spaces on homogeneous groups, in Harminic Analysis (Tianjin, 1998), Lecture Notes in Math. Vol., 1984 (Springer, Berlin, 1991), pp. 113–118.
doi: 10.1007/BFb0087762. |
[36] |
J. Liu, F. Weisz, D. Yang and W. Yuan, Littlewood-Paley and finite atomic characterizations of anisotropic variable Hardy-Lorentz spaces and their applications, J. Fourier Anal. Appl., (2018).
doi: 10.1007/s00041-018-9609-3. |
[37] |
J. Liu, D. Yang and W. Yuan,
Anisotropic Hardy-Lorentz spaces and their applications, Sci. China Math., 59 (2016), 1669-1720.
doi: 10.1007/s11425-016-5157-y. |
[38] |
J. Liu, D. Yang and W. Yuan,
Anisotropic variable Hardy-Lorentz spaces and their real interpolation, J. Math. Anal. Appl., 456 (2017), 356-393.
doi: 10.1016/j.jmaa.2017.07.003. |
[39] |
J. Musielak, Orlicz Spaces and Modular Spaces, Springer-Verlag, Berlin-NewYork, 1983.
doi: 10.1007/BFb0072210. |
[40] |
W. Orlicz,
Über eine gewisse Klasse von Raumen vom Typus B (German), Bull. Int. Acad. Pol. Ser. A, 8 (1932), 207-220.
|
[41] |
T. Quek and D. Yang,
Calderón-Zygmund-type operators on weighted weak Hardy spaces over ${{\mathbb R}^n}$, Acta Math. Sin., 16 (2000), 141-160.
doi: 10.1007/s101149900022. |
[42] |
E. M. Stein, M. Taibleson and G. Weiss,
Weak type estimates for maximal operators on certain Hp classes, Rend. Circ. Mat. Palermo, 1 (1981), 81-97.
|
[43] |
J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Berlin-New York: Springer-Verlag, 1989.
doi: 10.1007/BFb0091154. |
[44] |
M. H. Taibleson and G. Weiss, The molecular characterization of certain Hardy spaces, Asterisque, Mathematical Reviews, 77 (1980), 67–149. |
[45] |
L. Tang,
The weighted weak local Hardy spaces, Rocky Mountain J. Math., 44 (2014), 297-315.
doi: 10.1216/RMJ-2014-44-1-297. |
[46] |
H. Wang, Boundedness of several integral operators with bounded variable kernels on Hardy and weak Hardy spaces, Internat. J. Math., 24 (2013), 1350095, 22 pp.
doi: 10.1142/S0129167X1350095X. |
[47] |
D. Yang and S. Yang,
Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated to non-negative self-adjoint operators satisfying Gaussian estimates, Commun. Pure Appl. Anal., 15 (2016), 2135-2160.
doi: 10.3934/cpaa.2016031. |
[48] |
D. Yang, Y. Liang and L. D. Ky, Real-Variable Theory of Musielak-Orlicz Hardy Spaces, Lecture Notes in Mathematics, 2182, Springer-Verlag, Cham, 2017.
doi: 10.1007/978-3-319-54361-1. |
[49] |
H. Zhang, C. Qi and B. Li,
Anisotropic weak Hardy spaces of Musielak-Orlicz type and their applications, Front. Math. China, 12 (2017), 993-1022.
doi: 10.1007/s11464-016-0546-7. |
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