Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-10T16:07:50.876Z Has data issue: false hasContentIssue false
  • English
  • Français

MIXED NORM INEQUALITIES FOR SOME DIRECTIONAL MAXIMAL OPERATORS

Part of: Integral, integro-differential, and pseudodifferential operators Harmonic analysis in several variables

Published online by Cambridge University Press:  16 February 2012

DAH-CHIN LUOR
DAH-CHIN LUOR*
Affiliation:
Department of Applied Mathematics, I-Shou University, Dashu District, Kaohsiung City 84001, Taiwan (email: dclour@isu.edu.tw)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Mixed norm inequalities for directional operators are closely related to the boundedness problems of several important operators in harmonic analysis. In this paper we prove weighted inequalities for some one-dimensional one-sided maximal functions. Then by applying these results, we establish mixed norm inequalities for directional maximal operators which are defined from these one-dimensional maximal functions. We also estimate the constants in these inequalities.

Keywords

weighted inequalitiesmixed norm inequalitiesdirectional maximal operatorsmaximal functions

MSC classification

Secondary: 42B25: Maximal functions, Littlewood-Paley theory 47G10: Integral operators
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 3 , December 2012 , pp. 448 - 455
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

References

[1]Bez, N., ‘Mixed-norm estimates for a class of nonisotropic directional maximal operators and Hilbert transforms’, J. Funct. Anal. 255 (2008), 3281–3302.CrossRefGoogle Scholar
[2]Calderón, A. P. and Zygmund, A., ‘On singular integrals’, Amer. J. Math. 78 (1956), 289–309.Google Scholar
[3]Calderón, A. P. and Zygmund, A., ‘On singular integrals with variable kernel’, Appl. Anal. 7 (1978), 221–238.Google Scholar
[4]Chen, L.-K., ‘The singular integrals related to the Calderón–Zygmund method of rotations’, Appl. Anal. 30 (1988), 319–329.Google Scholar
[5]Chen, L.-K., ‘The maximal operators related to the Calderón–Zygmund method of rotations’, Illinois J. Math. 33 (1989), 268–279.Google Scholar
[6]Christ, M., Duoandikoetxea, J. and Rubio de Francia, J. L., ‘Maximal operators related to the Radon transform and the Calderón–Zygmund method of rotations’, Duke Math. J. 53 (1986), 189–209.Google Scholar
[7]Cowling, M. and Mauceri, G., ‘Inequalities for some maximal functions, I’, Trans. Amer. Math. Soc. 287 (1985), 431–455.Google Scholar
[8]Duoandikoetxea, J., ‘Weighted norm inequalities for homogeneous singular integrals’, Trans. Amer. Math. Soc. 336 (1993), 869–880.Google Scholar
[9]Duoandikoetxea, J., Fourier Analysis, Graduate Student in Mathematics, 29 (American Mathematical Society, Providence, RI, 2001).Google Scholar
[10]Duoandikoetxea, J., ‘Directional operators and mixed norms’, Publ. Mat. Special Issue (2002), 39–56 (Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations).CrossRefGoogle Scholar
[11]Duoandikoetxea, J. and Oruetxebarria, O., ‘Mixed norm inequalities for directional operators associated to potentials’, Potential Anal. 15 (2001), 273–283.Google Scholar
[12]Fefferman, R., ‘On an operator arising in the Calderón–Zygmund method of rotations and the Bramble–Hilbert lemma’, Proc. Natl. Acad. Sci. USA 80 (1983), 3877–3878.Google Scholar
[13]Hewitt, E. and Stromberg, K., Real and Abstract Analysis (Springer, Berlin, 1965).Google Scholar