Abstract
It is well-known that the pseudodifferential operator with the symbol in Bony class, a subset of \(S_{1,1}^0(\mathbb R^n)\), is bounded on \(L^{2}(\mathbb {R}^{n})\). The main purpose of this paper is to extend the classical results to multi-parameter case, i.e., to discuss the boundedness on \(L^2(\mathbb {R}^{n_1+n_2})\) and on \(h^{p}(\mathbb {R}^{n_{1}}\times \mathbb {R}^{n_{2}}) (0<p\le 1)\) of multi-parameter pseudodifferential operator with symbol satisfying multi-parameter Bony conditions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Álvarez, J., Hounie, J.: Estimates for the kernel and continuity properties of pseudodifferential operators. Ark. Mat. 28, 1–22 (1990)
Bony, J.M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. Ec. Norm. Super. 14, 209–246 (1981)
Bony, J.M.: Analyse microlocale des équations aux dérivées partielles non linéaires. In: Springer Lecture Notes in Math., vol. 1495, pp. 1–45. Springer, Berlin (1991)
Bui, H.: Weighted Hardy spaces. Math. Nachr. 103, 45–62 (1981)
Chang, S.-Y.A.: Carleson measures on the bi-disc. Ann. Math. 109, 613–620 (1979)
Chen, J., Huang, L., Lu, G.: Bi-parameter and bilinear Calderón̈CVaillancourt theorem with critical order. Forum Math. (2023). https://doi.org/10.1515/forum-2023-0458
Ching, C.-H.: Pseudodifferential operators with non regular symbols. J. Differ. Equ. 11, 436–447 (1972)
Chen, J., Ding, W., Lu, G.: Pseudodifferential operators on multi-parameter local hardy spaces. Forum Math. 32(4), 91–936 (2020)
Dai, W., Lu, G.: \(L^{p}\) estimates for multi-linear and multi-parameter pseudo-differential operators. Bull. Math. Sci. France 143(3), 567–597 (2015)
Ding, W., Ding X., Lu, G.: Continuity properties of multi-pseudodifferential operators. (Submitted)
Ding, W., Lu, G., Zhu, Y.: Multi-parameter local Hardy spaces. Nonlinear Anal. 184, 352–380 (2019)
Ding, W., Lu, G.: The boundedness of pseudodifferential operators on weighted multi-parameter local Hardy spaces via Fefferman’s criterion. Forum Math. 34(6), 1679–1705 (2022)
Ding, W., Lu, G., Zhu, Y.: Multi-parameter Triebel–Lizorkin spaces associated with the composition of two singular integrals and their atomic decomposition. Forum Math. 28, 25–42 (2016)
Ding, W., Lu, G., Zhu, Y.: Discrete Littlewood–Paley characterization of multi parameter local hardy spaces. Forum Math. 31(6), 1467–1488 (2019)
Goldberg, D.: A local version of real hardy spaces. Duke Math. J. 46(1), 27–42 (1979)
Grafakos, L.: Classical and Modern Fourier Analysis. Pearson, Upper Saddle River (2008)
Fefferman, R., Stein, E.M.: Singulsr integrals on product spaces. Adv. Math. 45, 117–143 (1982)
Fefferman, R.: Calderón–Zygmund theory for product domains-\(H^p\) spaces. Proc. Nat. Acad. Sci. 83, 840–843 (1986)
Fefferman, R.: Harmonic analysis on product spaces. Ann. Math. 126, 109–130 (1987)
Fefferman, R.: \(A^{p}\) weight and singular integrals. Am. J. Math. 110(5), 975–987 (1988)
Han, Y., Lin, C., Lu, G., Ruan, Z., Sawyer, E.: Hardy spaces associated with different homogeneities and boundedness of composition operators. Rev. Mat. Iberoam. 29(4), 1127–1157 (2013)
Han, Y., Lu, G., Ruan, Z.: Boundedness criterion of Journé’s class of singular integrals on multiparameter Hardy spaces. J. Funct. Anal. 264(5), 1238–1268 (2013)
Han, Y., Lu, G., Ruan, Z.: Boundedness of singular Integrals in Journé’s class on weighted multiparameter Hardy spaces. J. Geom. Anal. 24(4), 2186–2228 (2014)
Hörmander, L.: Pseudodifferential operators and hypoelliptic equations. Am. Math. Soc. Symp. Pure Math. 10, 138–183 (1967)
Hörmander, L.: On the continuity of pseudo-differential operators. Commun. Pure Appl. Math. 24, 529–535 (1971)
Hounie, J.: Rafael Augusto dos Santos Kapp, Pseudodifferential Operators on local hardy spaces. J. Fourier. Anal. Appl. 15, 153–178 (2009)
Huang, L., Chen, J.: The boundedness of multi-linear and multi-parameter pseudo-differential operators. Commun. Pure Appl. Anal. 20, 801–815 (2021)
Journé, J.L.: A cover lemma for product spaces. Proc. Am. Math. Soc. 96(4), 593–598 (1986)
Journé, J.L.: Calderman–Zygmund operators on product spaces. Rev. Mat. Iberoam. 1, 55–92 (1985)
Journé, J.L.: Two problems of Calderón–Zygmund theory on product spaces. Ann. Inst. Fourier (Grenoble) 38, 111–132 (1988)
Lu, G., Shen, J., Zhang, L.: Multi-parameter Hardy spaces theory and endpoint estimates for multi-parameter singular integrals. Mem. Amer. Math. Soc., 281, no. 1388, vi+87 pp (2023)
Muscalu, C., Pipher, J., Tao, T., et al.: Bi-parameter paraproducts. Acta Math. 193(2), 269–296 (2004)
Muscalu, C., Pipher, J., Tao, T., et al.: Multi-parameter paraproducts. Rev. Mat. Iberoam. 22(3), 963–976 (2006)
Pipher, J.: Journe’s covering lemma and its extension to higher dimensions. Duke Math. J. 53, 683–690 (1986)
Stein, E.: Harmonic Analysis, Real-Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton (1993)
Tang, L.: Weighted local Hardy spaces and their applications. Ill. J. Math. 56(2), 453–495 (2012)
Funding
National Natural Science Foundation of China (No.12271322 and 12271501)
Author information
Authors and Affiliations
Contributions
All authors contributed equally to this article. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supported by NNSF of China Grants (12271322, 12271501).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ding, W., Gu, M. & Zhu, Y. Continuity properties of multi-parameter pseudodifferential operators on Bony class. J. Pseudo-Differ. Oper. Appl. 15, 50 (2024). https://doi.org/10.1007/s11868-024-00622-1
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11868-024-00622-1