Two problems of Calderón-Zygmund theory on product-spaces
Annales de l'Institut Fourier, Tome 38 (1988) no. 1, pp. 111-132.

R. Fefferman a montré que sur un espace-produit à deux facteurs un opérateur T borné sur L2 est également borné de L dans BMO du produit si l’oscillation moyenne sur un rectangle R de l’image d’une fonction bornée supportée en dehors d’un multiple R de R est dominée par C|R|s|R|-s pour un s>0. Nous montrons ici que ce résultat n’est plus vrai en général pour un produit E de trois facteurs ou plus mais s’étend à ce cas lorsque l’opérateur T est un opérateur de convolution et s>s0(E). Également nous montrons que les bicommutateurs de Calderón-Coifman, obtenus à partir des commutateurs de Calderón par produit tensoriel multilinéaire, sont bornés sur L2 avec une croissance de norme polynomiale.

R. Fefferman has shown that, on a product-space with two factors, an operator T bounded on L2 maps L into BMO of the product if the mean oscillation on a rectangle R of the image of a bounded function supported out of a multiple R’ of R, is dominated by C|R|s|R|-s, for some s>0. We show that this result does not extend in general to the case where E has three or more factors but remains true in this case if in addition T is a convolution operator, provided s>s0(E). We also show that the Calderon-Coifman bicommutators, obtained from the Calderon commutators by multilinear tensor product, are bounded on L2 with norms growing polynomially.

@article{AIF_1988__38_1_111_0,
     author = {Journ\'e, Jean-Lin},
     title = {Two problems of {Calder\'on-Zygmund} theory on product-spaces},
     journal = {Annales de l'Institut Fourier},
     pages = {111--132},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {38},
     number = {1},
     year = {1988},
     doi = {10.5802/aif.1125},
     mrnumber = {90b:42031},
     zbl = {0638.47026},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1125/}
}
TY  - JOUR
AU  - Journé, Jean-Lin
TI  - Two problems of Calderón-Zygmund theory on product-spaces
JO  - Annales de l'Institut Fourier
PY  - 1988
SP  - 111
EP  - 132
VL  - 38
IS  - 1
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.1125/
DO  - 10.5802/aif.1125
LA  - en
ID  - AIF_1988__38_1_111_0
ER  - 
%0 Journal Article
%A Journé, Jean-Lin
%T Two problems of Calderón-Zygmund theory on product-spaces
%J Annales de l'Institut Fourier
%D 1988
%P 111-132
%V 38
%N 1
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.1125/
%R 10.5802/aif.1125
%G en
%F AIF_1988__38_1_111_0
Journé, Jean-Lin. Two problems of Calderón-Zygmund theory on product-spaces. Annales de l'Institut Fourier, Tome 38 (1988) no. 1, pp. 111-132. doi : 10.5802/aif.1125. http://www.numdam.org/articles/10.5802/aif.1125/

[1] R. Fefferman, Calderón-Zygmund theory for product domains-Hp spaces, P.N.S.A., vol. 83 (1986), 840-843. | MR | Zbl

[2] J. Pipher, Journé's covering lemma and its extension to higher dimensions, Duke Journal of Math, 53 n° 3 (1986), 683-690. | MR | Zbl

[3] J.-L. Journe, Calderón-Zygmund operators on product spaces, Mat. Iberoamerica, vol. 1, n° 3 (1985), 55-91. | MR | Zbl

[4] R. Fefferman, Harmonic analysis on product spaces, Ann. of Math., (2), vol. 126 (1987), 109-130. | MR | Zbl

[5] R. Fefferman, Functions of bounded mean oscillation on the polydisc, Ann. of Math., (2), 10 (1979), 395-406. | MR | Zbl

[6] L. Carleson, A counterexample for measures bounded for Hp for the bi-disc, Mittag Leffer Report n° 7, 1974.

[7] S.-Y. A. Chang and R. Fefferman, A continuous version of the duality of H1 and BMO on the bi-disc, Ann. of Math., (2), 112 (1980), 179-201. | MR | Zbl

[8] J.-L. Rubio De Francia, A Littlewood-Paley inequality for arbitrary intervals, Revista Mat. Iberoamericana, vol. 1, n° 2 (1985). | MR | Zbl

[9] P. Krikeles, Ph. D. dissertation, Yale University, 1982.

[10] M. Christ, and J.-L. Journé, Polynomial growth estimates for multilinear singular integral operators, Acta Math., 159 (1987), 51-80. | MR | Zbl

[11] S.-Y. A. Chang, Carleson measure on the bi-disc, Ann. of Math., (2), 109 (1979), 613-620. | MR | Zbl

[12] J.-L. Journé, Calderón-Zygmund operators, pseudo-differential operators and the Cauchy integral of Calderón, L.N. 994, Springer-Verlag. | MR | Zbl

[13] S.-Y. A. Chang and R. Fefferman, The Calderón-Zygmund decomposition on product domains, Am. J. of Math., vol. 104, 3, 455-468. | MR | Zbl

  • Cao, Yannan; Chang, Der-Chen; Wu, Xinfeng Relations between product and flag Triebel-Lizorkin spaces, Applicable Analysis, Volume 103 (2024) no. 14, p. 2516 | DOI:10.1080/00036811.2024.2302092
  • Ding, Wei; Gu, Min; Zhu, Yueping Continuity properties of multi-parameter pseudodifferential operators on Bony class, Journal of Pseudo-Differential Operators and Applications, Volume 15 (2024) no. 3, p. 31 (Id/No 50) | DOI:10.1007/s11868-024-00622-1 | Zbl:7901422
  • Chen, Jiao; He, Danqing; Lu, Guozhen; Park, Bae Jun; Zhang, Lu A sharp Hörmander estimate for multi-parameter and multi-linear Fourier multiplier operators, Mathematische Annalen, Volume 390 (2024) no. 4, pp. 5923-5985 | DOI:10.1007/s00208-024-02893-x | Zbl:7940276
  • Ding, Wei; Xu, Zheyuan; Zhu, YuePing Continuity of multi-parameter paraproduct, The Journal of Geometric Analysis, Volume 34 (2024) no. 8, p. 32 (Id/No 232) | DOI:10.1007/s12220-024-01669-8 | Zbl:1540.42038
  • Ding, Wei; Zou, Fangli Atomic decomposition of weighted multi-parameter mixed Hardy spaces, Bulletin of the Malaysian Mathematical Sciences Society. Second Series, Volume 46 (2023) no. 6, p. 20 (Id/No 182) | DOI:10.1007/s40840-023-01576-1 | Zbl:1527.42036
  • Lu, Guozhen; Shen, Jiawei; Zhang, Lu Multi-parameter Hardy spaces theory and endpoint estimates for multi-parameter singular integrals, Memoirs of the American Mathematical Society, 1388, Providence, RI: American Mathematical Society (AMS), 2023 | DOI:10.1090/memo/1388 | Zbl:1510.42002
  • Krantz, Steven Developments Since 1974, The E. M. Stein Lectures on Hardy Spaces, Volume 2326 (2023), p. 227 | DOI:10.1007/978-3-031-21952-8_6
  • Ding, Wei; Xu, Yun; Zhu, Yueping Weighted multi-parameter mixed Hardy spaces and their applications., Czechoslovak Mathematical Journal, Volume 72 (2022) no. 3, pp. 709-734 | DOI:10.21136/cmj.2022.0115-21 | Zbl:7584097
  • Ding, Wei; Lu, Guozhen Fefferman type criterion on weighted bi-parameter local Hardy spaces and boundedness of bi-parameter pseudodifferential operators, Forum Mathematicum, Volume 34 (2022) no. 6, pp. 1679-1705 | DOI:10.1515/forum-2022-0192 | Zbl:1503.42020
  • Xu, Chengdan A remark related pseudo-differential operators on multi-parameter Besov-Lipschitz and Triebel-Lizorkin spaces, Journal of Function Spaces, Volume 2022 (2022), p. 13 (Id/No 7953457) | DOI:10.1155/2022/7953457 | Zbl:7697662
  • Mozolyako, Pavel; Psaromiligkos, Georgios; Volberg, Alexander; Zorin-Kranich, Pavel Carleson embedding on the tri-tree and on the tri-disc, Revista Matemática Iberoamericana, Volume 38 (2022) no. 7, pp. 2069-2116 | DOI:10.4171/rmi/1378 | Zbl:1510.42032
  • Chen, Jiao; Huang, Liang; Lu, Guozhen Hörmander Fourier multiplier theorems with optimal Besov regularity on multi-parameter Hardy spaces, Forum Mathematicum, Volume 33 (2021) no. 6, pp. 1605-1627 | DOI:10.1515/forum-2021-0201 | Zbl:1486.42011
  • Ding, Wei; Yu, Feng Dual spaces of multiparameter local Hardy spaces, Journal of Function Spaces, Volume 2021 (2021), p. 12 (Id/No 9619925) | DOI:10.1155/2021/9619925 | Zbl:1480.42032
  • Chang, Der-Chen; Han, Yongsheng; Wu, Xinfeng Relations between product and flag Hardy spaces, The Journal of Geometric Analysis, Volume 31 (2021) no. 7, pp. 6601-6623 | DOI:10.1007/s12220-019-00284-2 | Zbl:1471.42034
  • Li, Kangwei; Martell, José María; Martikainen, Henri; Ombrosi, Sheldy; Vuorinen, Emil End-point estimates, extrapolation for multilinear Muckenhoupt classes, and applications, Transactions of the American Mathematical Society, Volume 374 (2021) no. 1, pp. 97-135 | DOI:10.1090/tran/8172 | Zbl:1467.42022
  • Ding, Wei; Zhu, Yueping Mixed Hardy spaces and their applications, Acta Mathematica Scientia. Series B. (English Edition), Volume 40 (2020) no. 4, pp. 945-969 | DOI:10.1007/s10473-020-0405-1 | Zbl:1499.42110
  • Huang, Liang; Chen, Jiao Hörmander type multiplier theorems on bi-parameter anisotropic Hardy spaces, Forum Mathematicum, Volume 32 (2020) no. 3, pp. 577-594 | DOI:10.1515/forum-2019-0268 | Zbl:1439.42012
  • Chen, Jiao; Ding, Wei; Lu, Guozhen Boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces, Forum Mathematicum, Volume 32 (2020) no. 4, pp. 919-936 | DOI:10.1515/forum-2019-0319 | Zbl:1442.42054
  • Li, Kangwei; Martikainen, Henri; Vuorinen, Emil Bilinear Calderón-Zygmund theory on product spaces, Journal de Mathématiques Pures et Appliquées. Neuvième Série, Volume 138 (2020), pp. 356-412 | DOI:10.1016/j.matpur.2019.10.007 | Zbl:1439.42020
  • Ding, Wei; Qin, Meidi; Zhu, Yueping The boundedness on mixed Hardy spaces, Journal of Function Spaces, Volume 2020 (2020), p. 12 (Id/No 5341674) | DOI:10.1155/2020/5341674 | Zbl:1436.42018
  • Ding, Wei; Lu, Guozhen Boundedness of inhomogeneous Journé's type operators on multi-parameter local Hardy spaces, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 197 (2020), p. 30 (Id/No 111816) | DOI:10.1016/j.na.2020.111816 | Zbl:1440.42052
  • Han, Yanchang; Han, Yongsheng; Li, Ji; Tan, Chaoqiang Marcinkiewicz multipliers and Lipschitz spaces on Heisenberg groups, Canadian Journal of Mathematics, Volume 71 (2019) no. 3, pp. 607-627 | DOI:10.4153/cjm-2018-003-0 | Zbl:1418.42029
  • Hong, Qing; Hu, Guorong Molecular decomposition and a class of Fourier multipliers for bi-parameter modulation spaces, Communications on Pure and Applied Analysis, Volume 18 (2019) no. 6, pp. 3103-3120 | DOI:10.3934/cpaa.2019139 | Zbl:1480.42035
  • Ding, Wei; Chen, Jiao; Niu, Yaoming Note on duality of weighted multi-parameter Triebel-Lizorkin spaces., Czechoslovak Mathematical Journal, Volume 69 (2019) no. 3, pp. 763-779 | DOI:10.21136/cmj.2019.0509-17 | Zbl:1513.42076
  • Ding, Wei; Lu, Guozhen; Zhu, Yueping Discrete Littlewood-Paley-Stein characterization of multi-parameter local Hardy spaces, Forum Mathematicum, Volume 31 (2019) no. 6, pp. 1467-1488 | DOI:10.1515/forum-2019-0038 | Zbl:1440.42104
  • Grafakos, Loukas; He, Danqing; Nguyen, Hanh Van; Yan, Lixin Multilinear multiplier theorems and applications, The Journal of Fourier Analysis and Applications, Volume 25 (2019) no. 3, pp. 959-994 | DOI:10.1007/s00041-018-9606-6 | Zbl:1423.42018
  • Xu, Chengdan; Huang, Liang Boundedness of bi-parameter pseudo-differential operators on bi-parameter α-modulation spaces, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 180 (2019), pp. 20-40 | DOI:10.1016/j.na.2018.09.004 | Zbl:1411.35296
  • Ding, Wei; Lu, Guozhen; Zhu, YuePing Multi-parameter local Hardy spaces, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 184 (2019), pp. 352-380 | DOI:10.1016/j.na.2019.02.014 | Zbl:1418.42038
  • Han, Yongsheng; Lin, Chin-Cheng; Wu, Xinfeng Boundedness of singular integrals with flag kernels on weighted flag Hardy spaces, Pacific Journal of Mathematics, Volume 302 (2019) no. 2, pp. 545-598 | DOI:10.2140/pjm.2019.302.545 | Zbl:1434.42017
  • Brummer, Joshua; Naibo, Virginia Weighted fractional Leibniz-type rules for bilinear multiplier operators, Potential Analysis, Volume 51 (2019) no. 1, pp. 71-99 | DOI:10.1007/s11118-018-9703-9 | Zbl:1418.42024
  • Han, Yongsheng; Li, Ji; Lin, Chin-Cheng; Tan, Chaoqiang Singular integrals associated with Zygmund dilations, The Journal of Geometric Analysis, Volume 29 (2019) no. 3, pp. 2410-2455 | DOI:10.1007/s12220-018-0081-8 | Zbl:1430.42017
  • Chen, Jiao; Lu, Guozhen Hörmander type theorem on bi-parameter Hardy spaces for bi-parameter Fourier multipliers with optimal smoothness, Revista Matemática Iberoamericana, Volume 34 (2018) no. 4, pp. 1541-1561 | DOI:10.4171/rmi/1035 | Zbl:1422.42023
  • DING, Wei; ZHU, Yueping Duality of weighted multiparameter Triebel-Lizorkin spaces, Acta Mathematica Scientia, Volume 37 (2017) no. 4, p. 1083 | DOI:10.1016/s0252-9602(17)30059-0
  • Chen, Jiao Hörmander type theorem for Fourier multipliers with optimal smoothness on Hardy spaces of arbitrary number of parameters, Acta Mathematica Sinica. English Series, Volume 33 (2017) no. 8, pp. 1083-1106 | DOI:10.1007/s10114-017-6526-3 | Zbl:1373.42011
  • Weisz, Ferenc Multi-Dimensional Hardy Spaces, Convergence and Summability of Fourier Transforms and Hardy Spaces (2017), p. 137 | DOI:10.1007/978-3-319-56814-0_3
  • Weisz, Ferenc Multi-Dimensional Fourier Transforms, Convergence and Summability of Fourier Transforms and Hardy Spaces (2017), p. 203 | DOI:10.1007/978-3-319-56814-0_4
  • Weisz, Ferenc ℓ q -Summability of Multi-Dimensional Fourier Transforms, Convergence and Summability of Fourier Transforms and Hardy Spaces (2017), p. 229 | DOI:10.1007/978-3-319-56814-0_5
  • Weisz, Ferenc One-Dimensional Hardy Spaces, Convergence and Summability of Fourier Transforms and Hardy Spaces (2017), p. 3 | DOI:10.1007/978-3-319-56814-0_1
  • Weisz, Ferenc Rectangular Summability of Multi-Dimensional Fourier Transforms, Convergence and Summability of Fourier Transforms and Hardy Spaces (2017), p. 383 | DOI:10.1007/978-3-319-56814-0_6
  • Weisz, Ferenc One-Dimensional Fourier Transforms, Convergence and Summability of Fourier Transforms and Hardy Spaces (2017), p. 71 | DOI:10.1007/978-3-319-56814-0_2
  • Chen, Lu; Lu, Guozhen; Luo, Xiang Boundedness of multi-parameter Fourier multiplier operators on Triebel-Lizorkin and Besov-Lipschitz spaces, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 134 (2016), pp. 55-69 | DOI:10.1016/j.na.2015.12.016 | Zbl:1333.42013
  • Ding, Wei; Lu, Guozhen Duality of multi-parameter Triebel-Lizorkin spaces associated with the composition of two singular integral operators, Transactions of the American Mathematical Society, Volume 368 (2016) no. 10, pp. 7119-7152 | DOI:10.1090/tran/6576 | Zbl:1338.42025
  • Cabral, Raquel Differentiation of the integral, Hardy spaces and Calderón-Zygmund operators in the product setting, Journal of Mathematical Analysis and Applications, Volume 423 (2015) no. 2, pp. 1480-1513 | DOI:10.1016/j.jmaa.2014.10.047 | Zbl:1304.42054
  • Ding, Wei The boundedness of composition operators on Triebel-Lizorkin and Besov spaces with different homogeneities, Acta Mathematica Sinica. English Series, Volume 30 (2014) no. 6, pp. 933-948 | DOI:10.1007/s10114-014-3280-7 | Zbl:1433.42008
  • Han, Yongsheng; Lu, Guozhen; Sawyer, Eric Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group, Analysis PDE, Volume 7 (2014) no. 7, p. 1465 | DOI:10.2140/apde.2014.7.1465
  • Muscalu, Camil Calderón commutators and the Cauchy integral on Lipschitz curves revisited III. Polydisc extensions, Revista Matemática Iberoamericana, Volume 30 (2014) no. 4, pp. 1413-1437 | DOI:10.4171/rmi/820 | Zbl:1311.42042
  • Han, Yongsheng; Lu, Guozhen; Ruan, Zhuoping Boundedness of singular integrals in Journé's class on weighted multiparameter Hardy spaces, The Journal of Geometric Analysis, Volume 24 (2014) no. 4, pp. 2186-2228 | DOI:10.1007/s12220-013-9421-x | Zbl:1302.42024
  • Lu, Guo Zhen; Zhu, Yue Ping Singular integrals and weighted Triebel-Lizorkin and Besov spaces of arbitrary number of parameters, Acta Mathematica Sinica. English Series, Volume 29 (2013) no. 1, pp. 39-52 | DOI:10.1007/s10114-012-1402-7 | Zbl:1261.42030
  • Han, Yongsheng; Lu, Guozhen; Ruan, Zhuoping Boundedness criterion of Journé's class of singular integrals on multiparameter Hardy spaces, Journal of Functional Analysis, Volume 264 (2013) no. 5, pp. 1238-1268 | DOI:10.1016/j.jfa.2012.12.006 | Zbl:1268.42024
  • Han, Yongsheng; Li, Ji; Lu, Guozhen Multiparameter Hardy space theory on Carnot-Carathéodory spaces and product spaces of homogeneous type, Transactions of the American Mathematical Society, Volume 365 (2013) no. 1, pp. 319-360 | DOI:10.1090/s0002-9947-2012-05638-8 | Zbl:1275.42035
  • Zhu, Xiangrong; Chen, Jiecheng Integrability of the general product Hardy operators on the product Hardy spaces, Applied Mathematics. Series B (English Edition), Volume 27 (2012) no. 2, pp. 225-233 | DOI:10.1007/s11766-012-2973-x | Zbl:1265.42086
  • Li, Ji; Song, Liang; Tan, Chaoqiang Various characterizations of product Hardy space, Proceedings of the American Mathematical Society, Volume 139 (2011) no. 12, pp. 4385-4400 | DOI:10.1090/s0002-9939-2011-10852-9 | Zbl:1252.42022
  • Weisz, Ferenc Applications of multi-parameter martingales in Fourier analysis, Stochastics and Dynamics, Volume 11 (2011) no. 2-3, pp. 551-568 | DOI:10.1142/s0219493711003449 | Zbl:1231.42028
  • Ruan, Zhuoping Non-Isotropic Flag Singular Integrals on Multi-Parameter Hardy Spaces, Taiwanese Journal of Mathematics, Volume 15 (2011) no. 2 | DOI:10.11650/twjm/1500406217
  • Ding, Yong; Lu, Guo Zhen; Ma, Bo Lin Multi-parameter Triebel-Lizorkin and Besov spaces associated with flag singular integrals, Acta Mathematica Sinica. English Series, Volume 26 (2010) no. 4, pp. 603-620 | DOI:10.1007/s10114-010-8352-8 | Zbl:1192.42014
  • Ruan, Zhuoping Multi-parameter Hardy spaces via discrete Littlewood-Paley theory, Analysis in Theory and Applications, Volume 26 (2010) no. 2, pp. 122-139 | DOI:10.1007/s10496-010-0122-z | Zbl:1224.42039
  • Ruan, Zhuoping Weighted Hardy spaces in the three-parameter case, Journal of Mathematical Analysis and Applications, Volume 367 (2010) no. 2, pp. 625-639 | DOI:10.1016/j.jmaa.2010.02.010 | Zbl:1198.42015
  • Kenig, Carlos E. On the local and global well-posedness theory for the KP-I equation, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, Volume 21 (2004) no. 6, pp. 827-838 | DOI:10.1016/j.anihpc.2003.12.002 | Zbl:1072.35162
  • Carbery, Anthony; Seeger, Andreas 𝐻^𝑝- and 𝐿^𝑝-variants of multiparameter Calderón-Zygmund theory, Transactions of the American Mathematical Society, Volume 334 (1992) no. 2, p. 719 | DOI:10.1090/s0002-9947-1992-1072104-4
  • Fefferman, Robert Some recent developments in Fourier analysis and HP theory and product domains. II, Function spaces and applications, Proc. US-Swed. Semin., Lund/Swed., Lect. Notes Math. 1302, 44-51, 1988 | Zbl:0658.42023

Cité par 60 documents. Sources : Crossref, zbMATH