Abstract
This work explores new deep connections between John–Nirenberg type inequalities and Muckenhoupt weight invariance for a large class of BMO-type spaces. The results are formulated in a very general framework in which BMO spaces are constructed using a base of sets, used also to define weights with respect to a non-negative measure (not necessarily doubling), and an appropriate oscillation functional. This includes as particular cases many different function spaces on geometric settings of interest. As a consequence, the weight invariance of several BMO and Triebel–Lizorkin spaces considered in the literature is proved. Most of the invariance results obtained under this unifying approach are new even in the most classical settings.
Similar content being viewed by others
References
Bernicot, F., Martell, J.M.: Self-improving properties for abstract Poincaré type inequalities. Trans. Am. Math. Soc. 367(7), 4793–4835 (2015)
Bernicot, F., Zhao, J.: Abstract framework for John Nirenberg inequalities and applications to Hardy spaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11(3), 475–501 (2012)
Bloom, S.: A commutator theorem and weighted BMO. Trans. Am. Math. Soc. 292(1), 103–122 (1985)
Bui, H.-Q., Duong, X.-T.: Weighted BMO spaces associated to operators, preprint. arXiv:1201.5828v3 [math.FA] (2013)
Bui, H.-Q., Taibleson, M.: The characterization of the Triebel-Lizorkin spaces for \(p=\infty \). J. Fourier Anal. Appl. 6(5), 537–550 (2000)
Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83(4), 569–645 (1977)
Duong, X.T., Yan, L.: Duality of Hardy and BMO spaces associated with operators with heat kernel bounds. J. Am. Math. Soc. 18(4), 943–973 (2005)
Duong, X.T., Yan, L.: New function spaces of BMO type, the John-Nirenberg inequality, interpolation, and applications. Commun. Pure Appl. Math. 58(10), 1375–1420 (2005)
Franchi, B., Pérez, C., Wheeden, R.: Self-improving properties of John-Nirenberg and Poincaré inequalities on spaces of homogeneous type. J. Funct. Anal. 153(5), 108–146 (1998)
Frazier, M., Jawerth, B.: A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93(1), 34–170 (1990)
García-Cuerva, J.: Weighted \(H^p\) spaces. Diss. Math. (Rozprawy Mat.) 162, 63 (1979)
García-Cuerva, J., Rubio de Francia, J.-L.: Weighted Norm Inequalities and Related Topics, North-Holland Mathematics Studies, 116, Notas de Matemática, 104. North-Holland Publishing Co., Amsterdam (1985)
Harboure, E., Salinas, O., Viviani, B.: A look at \(BMO_\varphi (\omega )\) through Carleson measures. J. Fourier Anal. Appl. 13(3), 267–284 (2007)
Hartzstein, S., Salinas, O.: Weighted BMO and Carleson measures on spaces of homogeneous type. J. Math. Anal. Appl. 342(2), 950–969 (2008)
Hart, J., Oliveira, L.: Hardy space estimates for limited ranges of Muckenhoupt weights. Adv. Math. 313, 803–838 (2017)
Ho, K.-P.: Characterizations of \(BMO\) by \(A_p\) weights and \(p\)-convexity. Hiroshima Math. J. 41(2), 153–165 (2011)
Hofmann, S., Mayboroda, S.: Hardy and BMO spaces associated to divergence form elliptic operators. Math. Ann. 344(1), 37–116 (2009)
Holmes, I., Lacey, M.T., Wick, W.D.: Commutators in the two-weight setting. Math. Ann. 367(1–2), 51–80 (2017)
Hytönen, T., Pérez, C.: Sharp weighted bounds involving \(A_\infty \). Anal. PDE 6(4), 777–818 (2013)
Hytönen, T., Pérez, C., Rela, E.: Sharp reverse Hölder property for \(A_\infty \) weights on spaces of homogeneous types. J. Funct. Anal. 263(12), 3883–3899 (2012)
Jiménez-del-Toro, A.: Exponential self-improvement of generalized Poincaré inequalities associated with approximations of the identity and semigroups. Trans. Am. Math. Soc. 364(2), 637–660 (2012)
Jiménez-del-Toro, A., Martell, J.M.: Self-improvement of Poincaré type inequalities associated with approximations of the identity and semigroups. Potential Anal. 38(3), 805–841 (2013)
John, F., Nirenberg, L.: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14, 415–426 (1961)
Journé, J.-L.: Calderón-Zygmund operators, pseudodifferential operators and the Cauchy integral of Calderón. Lecture Notes in Mathematics, vol. 994. Springer, Berlin (1983)
Lo, Y., Ruilin, L.: \(BMO\) functions in spaces of homogeneous type. Sci. Sin. Ser. A 27(7), 695–708 (1984)
Logunov, A.A., Slavin, L., Stolyarov, D.M., Vasyunin, V., Zatitskiy, P.B.: Weak integral conditions for BMO. Proc. Am. Math. Soc. 143(7), 2913–2926 (2015)
Luque, T., Pérez, C., Rela, E.: Reverse Hölder property for strong weights and general measures. J. Geom. Anal. 27(1), 162–182 (2017)
Mateu, J., Mattila, P., Nicolau, A., Orobitg, J.: BMO for nondoubling measures. Duke Math. J. 102(3), 533–565 (2000)
Muckenhoupt, B., Wheeden, R.: Weighted bounded mean oscillation and the Hilbert transform. Studia Math. 54(3), 221–237 (1975/76)
Neri, U.: Fractional integration on the space \(H^1\) and its dual. Studia Math. 53(2), 175–189 (1975)
Orobitg, J., Pérez, C.: \(A_p\) weights for nondoubling measures in \({\mathbb{R}}^n\) and applications. Trans. Am. Math. Soc. 354(5), 2013–2033 (2002)
Shi, X.L., Torchinsky, A.: Local sharp maximal functions in spaces of homogeneous type. Sci. Sin. Ser. A 30(5), 473–480 (1987)
Strichartz, R.: Bounded mean oscillation and Sobolev spaces. Indiana Univ. Math. J. 29(4), 539–558 (1980)
Strichartz, R.: Traces of BMO-Sobolev spaces. Proc. Am. Math. Soc. 83(3), 509–513 (1981)
Strömberg, J.-O.: Bounded mean oscillation with Orlicz norms and duality of Hardy spaces. Indiana Univ. Math, J. 28(3), 511–544 (1979)
Trong, N., Tung, N.: Weighted \(BMO\) type spaces associated to admissible functions and applications. Acta Math. Vietnam 41(2), 209–241 (2016)
Tsutsui, Y.: \(A_\infty \) constants between \(BMO\) and weighted \(BMO\). Proc. Jpn. Acad. Ser. A Math. Sci. 90(1), 11–14 (2014)
Acknowledgements
We would like to thank the anonymous referee for a thorough reading of our original manuscript and for the very valuable observations which have helped clarify and improve our presentation.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hart, J., Torres, R.H. John–Nirenberg Inequalities and Weight Invariant BMO Spaces. J Geom Anal 29, 1608–1648 (2019). https://doi.org/10.1007/s12220-018-0054-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-018-0054-y