Hostname: page-component-6bf8c574d5-86b6f Total loading time: 0 Render date: 2025-03-07T08:00:25.632Z Has data issue: false hasContentIssue false
  • English
  • Français

Dyadic John–Nirenberg space

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  17 November 2021

Juha Kinnunen  and
Kim Myyryläinen Open the ORCID record for Kim Myyryläinen [Opens in a new window]
Juha Kinnunen
Affiliation:
Department of Mathematics, Aalto University, P.O. Box 11100, FI-00076 Aalto, Finland ([email protected], [email protected])
Kim Myyryläinen
Affiliation:
Department of Mathematics, Aalto University, P.O. Box 11100, FI-00076 Aalto, Finland ([email protected], [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We discuss the dyadic John–Nirenberg space that is a generalization of functions of bounded mean oscillation. A John–Nirenberg inequality, which gives a weak type estimate for the oscillation of a function, is discussed in the setting of medians instead of integral averages. We show that the dyadic maximal operator is bounded on the dyadic John–Nirenberg space and provide a method to construct nontrivial functions in the dyadic John–Nirenberg space. Moreover, we prove that the John–Nirenberg space is complete. Several open problems are also discussed.

Keywords

John–Nirenberg spacedyadicmaximal functionmedianJohn–Nirenberg inequality

MSC classification

Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 42B35: Function spaces arising in harmonic analysis
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 1 , February 2023 , pp. 1 - 18
Check for updates
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aalto, D., Berkovits, L., Kansanen, O. E. and Yue, H.. John–Nirenberg lemmas for a doubling measure. Studia Math. 204 (2011), 21–37.CrossRefGoogle Scholar
Bennett, C., DeVore, R. A. and Sharpley, R.. Weak-$L^{\infty }$ and BMO. Ann. Math. (2) 113 (1981), 601–611.CrossRefGoogle Scholar
Berkovits, L., Kinnunen, J. and Martell, J. M.. Oscillation estimates, self-improving results and good-$\lambda$ inequalities. J. Funct. Anal. 270 (2016), 3559–3590.CrossRefGoogle Scholar
Chiarenza, F. and Frasca, M.. Morrey spaces and Hardy–Littlewood maximal function. Rend. Mat. Appl. (7) 7 (1987), 273–279 (1988).Google Scholar
Coifman, R. R. and Rochberg, R.. Another characterization of BMO. Proc. Amer. Math. Soc. 79 (1980), 249–254.CrossRefGoogle Scholar
Dafni, G., Hytönen, T., Korte, R. and Yue, H.. The space $JN_p$: nontriviality and duality. J. Funct. Anal. 275 (2018), 577–603.CrossRefGoogle Scholar
Federer, H. and Ziemer, W. P.. The Lebesgue set of a function whose distribution derivatives are $p$-th power summable. Indiana Univ. Math. J. 22 (1972/73), 139–158.CrossRefGoogle Scholar
Fujii, N.. A condition for a two-weight norm inequality for singular integral operators. Studia Math. 98 (1991), 175–190.CrossRefGoogle Scholar
Garnett, J. B. and Jones, P. W.. BMO from dyadic BMO. Pacific J. Math. 99 (1982), 351–371.CrossRefGoogle Scholar
Gogatishvili, A., Koskela, P. and Zhou, Y.. Characterizations of Besov and Triebel–Lizorkin spaces on metric measure spaces. Forum Math. 25 (2013), 787–819.Google Scholar
Heikkinen, T.. Generalized Lebesgue points for Hajłasz functions. J. Funct. Spaces (2018), Art. ID 5637042, 12 pp.CrossRefGoogle Scholar
Heikkinen, T., Ihnatsyeva, L. and Tuominen, H.. Measure density and extension of Besov and Triebel–Lizorkin functions. J. Fourier Anal. Appl. 22 (2016), 334–382.CrossRefGoogle Scholar
Heikkinen, T. and Kinnunen, J.. A median approach to differentiation bases. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 30 (2019), 41–66.CrossRefGoogle Scholar
Heikkinen, T., Koskela, P. and Tuominen, H.. Approximation and quasicontinuity of Besov and Triebel–Lizorkin functions. Trans. Amer. Math. Soc. 369 (2017), 3547–3573.CrossRefGoogle Scholar
Heikkinen, T. and Tuominen, H.. Approximation by Hölder functions in Besov and Triebel–Lizorkin spaces. Constr. Approx. 44 (2016), 455–482.CrossRefGoogle Scholar
Jawerth, B., Pérez, C. and Welland, G., The positive cone in Triebel–Lizorkin spaces and the relation among potential and maximal operators. In Harmonic analysis and partial differential equations (Boca Raton, FL, 1988), (eds M. Milman and T. Schonbek) Contemp. Math., vol. 107, pp. 71–91 (Providence, RI: Amer. Math. Soc., 1990).CrossRefGoogle Scholar
Jawerth, B. and Torchinsky, A.. Local sharp maximal functions. J. Approx. Theory 43 (1985), 231–270.CrossRefGoogle Scholar
John, F., Quasi-isometric mappings. In Seminari 1962/63 Anal. Alg. Geom. e Topol., vol. 2, pp. 462–473 (Ist. Naz. Alta Mat., Ediz. Cremonese, Rome, 1965).Google Scholar
John, F. and Nirenberg, L.. On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14 (1961), 415–426.CrossRefGoogle Scholar
Karak, N.. Triebel–Lizorkin capacity and Hausdorff measure in metric spaces. Math. Slovaca 70 (2020), 617–624.CrossRefGoogle Scholar
Lerner, A. K.. A pointwise estimate for the local sharp maximal function with applications to singular integrals. Bull. Lond. Math. Soc. 42 (2010), 843–856.CrossRefGoogle Scholar
Lerner, A. K. and Pérez, C.. Self-improving properties of generalized Poincaré type inequalities through rearrangements. Math. Scand. 97 (2005), 217–234.CrossRefGoogle Scholar
Myyryläinen, K.. Median-type John–Nirenberg space in metric measure spaces. Preprint, arXiv:2104.05380 (2021).CrossRefGoogle Scholar
Neri, U.. Some properties of functions with bounded mean oscillation. Studia Math. 61 (1977), 63–75.CrossRefGoogle Scholar
Poelhuis, J. and Torchinsky, A.. Medians, continuity, and vanishing oscillation. Studia Math. 213 (2012), 227–242.CrossRefGoogle Scholar
Stein, E. M.. Note on the class $L\log L$. Studia Math. 32 (1969), 305–310.CrossRefGoogle Scholar
Strömberg, J.-O.. Bounded mean oscillation with Orlicz norms and duality of Hardy spaces. Indiana Univ. Math. J. 28 (1979), 511–544.CrossRefGoogle Scholar
Strömberg, J.-O. and Torchinsky, A.. Weighted Hardy spaces. Lecture Notes in Mathematics, vol. 1381 (Berlin: Springer-Verlag, 1989).CrossRefGoogle Scholar
Zhou, Y.. Fractional Sobolev extension and imbedding. Trans. Amer. Math. Soc. 367 (2015), 959–979.CrossRefGoogle Scholar