Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-23T04:58:10.949Z Has data issue: false hasContentIssue false
  • English
  • Français

The one-sided Ap conditions and local maximal operator

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  07 November 2011

Ana L. Bernardis ,
Amiran Gogatishvili ,
Francisco Javier Martín-Reyes ,
Pedro Ortega Salvador  and
Luboš Pick
Ana L. Bernardis
Affiliation:
Instituto de Matemática Aplicada del Litoral (CONICET), Facultad de Ingeniería Química (UNL), Güemes 3450, (3000) Santa Fe, Argentina ([email protected])
Amiran Gogatishvili
Affiliation:
Mathematical Institute, Academy of Sciences of the Czech Republic, Žitná 25, 11567 Praha 1, Czech Republic ([email protected])
Francisco Javier Martín-Reyes
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain ([email protected]; [email protected])
Pedro Ortega Salvador
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain ([email protected]; [email protected])
Luboš Pick
Affiliation:
Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 18675 Praha 8, Czech Republic ([email protected])
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.

We introduce the one-sided local maximal operator and study its connection to the one-sided Ap conditions. We get a new characterization of the boundedness of the one-sided maximal operator on a quasi-Banach function space. We obtain applications to weighted Lebesgue spaces and variable-exponent Lebesgue spaces.

Keywords

one-sided Ap conditionsone-sided local maximal operatorquasi-Banach function spacesvariable-exponent Lebesgue spaces

MSC classification

Secondary: 42B25: Maximal functions, Littlewood-Paley theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 1 , February 2012 , pp. 79 - 104
Copyright
Copyright © Edinburgh Mathematical Society 2011

References

1.Bennett, C. and Sharpley, R., Interpolation of operators, Pure and Applied Mathematics, Volume 1299 (Academic Press, Boston, MA, 1988).Google Scholar
2.Bernardis, A. L. and Lorente, M., Sharp two weight inequalities for commutators of Riemann–Liouville and Weyl fractional integral operators, Integ. Eqns Operat. Theory 61(4) (2008), 449–475.CrossRefGoogle Scholar
3.Edmunds, D. E., Kokilashvili, V. and Meskhi, A., On one-sided operators in variable exponent Lebesgue spaces, Proc. A. Razmadze Math. Inst. 144 (2007), 126–131.Google Scholar
4.Hewitt, E. and Stromberg, K., Real and abstract analysis, Graduate Texts in Mathematics, Volume 25 (Springer, 1965).Google Scholar
5.Kalton, N. J., Peck, N. T. and Roberts, J. W., An F-space sampler, London Mathematical Society Lecture Notes Series, Volume 89 (Cambridge University Press, 1984).Google Scholar
6.Lerner, A. K. and Pérez, C., A new characterization of the Muckenhoupt Ap weights through an extension of the Lorentz–Shimogaki Theorem, Indiana Univ. Math. J. 56(6) (2007), 2697–2722.CrossRefGoogle Scholar
7.Martín-Reyes, F. J., New proofs of weighted inequalities for the one-sided Hardy–Littlewood maximal functions, Proc. Am. Math. Soc. 117 (1993), 691–698.CrossRefGoogle Scholar
8.Muckenhoupt, B., Weighted norm inequalities for the Hardy maximal function, Trans. Am. Math. Soc. 165 (1972), 31–38.CrossRefGoogle Scholar
9.Musiełak, J., Orlicz spaces and modular spaces, Lecture Notes in Mathematics, Volume 1034 (Springer, 1983).Google Scholar
10.Nekvinda, A., Hardy–Littlewood maximal operator in Lp(x) (ℝ), Math. Inequal. Applic. 7 (2004), 255–265.Google Scholar
11.Nekvinda, A., A note on one-sided maximal operator in Lp(·) (ℝ), Math. Inequal. Appl. 13(4) (2010), 887–897.Google Scholar
12.Salvador, P. Ortega, Weighted inequalities for one-sided maximal functions in Orlicz spaces, Studia Math. 131 (1998), 101–114.CrossRefGoogle Scholar
13.Sawyer, E., Weighted inequalities for the one-sided Hardy–Littlewood maximal functions, Trans. Am. Math. Soc. 297 (1986), 53–61.CrossRefGoogle Scholar
14.Stein, E. M., Note on the class L(log L), Studia Math. 32 (1969), 305–310.CrossRefGoogle Scholar