Abstract
This paper studies functions of bounded mean oscillation (\({{\mathrm{BMO}}}\)) on metric spaces equipped with a doubling measure. The main result gives characterizations for mappings that preserve \({{\mathrm{BMO}}}\). This extends the corresponding Euclidean results by Gotoh to metric measure spaces. The argument is based on a generalization Uchiyama’s construction of certain extremal \({{\mathrm{BMO}}}\)-functions and John-Nirenberg’s lemma.
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References
Astala, K.: A remark on quasiconformal mappings and BMO-functions. Michigan Math. J. 30, 209–212 (1983)
Björn, A., Björn, J.: Nonlinear Potential Theory on Metric Spaces, EMS Tracts in Mathematics, 17. European Mathematical Society (EMS), Zürich (2011)
Buckley, S.M.: Inequalities of John-Nirenberg type in doubling spaces. J. Anal. Math. 79, 215–240 (1999)
García-Cuerva, J., Rubio de Francia, J.L.: Weighted Norm Inequalities and Related Topics, North-Holland Mathematics Studies, 116. North-Holland Publishing Co., Amsterdam (1985)
Garnett, J.B., Jones, P.W.: The distance in BMO to \(L^\infty \). Ann. Math. 108(2), 373–393 (1978)
Gehring, F.W., Kelly, J.C.: Quasi-conformal mappings and Lebesgue density, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Mary- land, College Park, Md., 1973), pp. 171–179. Ann. Math. Stud., No. 79, Princeton Univ. Press, Princeton, N.J., (1974)
Gotoh, Y.: An extension of the Uchiyama theorem and its application to composition operators which preserve BMO. J. Anal. Math. 97, 133–167 (2006)
Gotoh, Y.: On composition operators which preserve BMO. Pac. J. Math. 201, 289–307 (2001)
Kinnunen, J., Korte, R., Shanmugalingam, N., Tuominen, H.: Lebesgue points and capacities via the boxing inequality in metric spaces. Indiana Univ. Math. J. 57, 401–430 (2008)
Korte, R., Kansanen, O.E.: Strong \(A_\infty \)- weights are \(A_\infty \)-weights on metric spaces. Rev. Mat. Iberoam. 27, 335–354 (2011)
Korte, R., Marola, N., Shanmugalingam, N.: Quasiconformality, homeomorphisms between metric measure spaces preserving quasiminimizers, and uniform density property. Ark. Mat. 50, 111–134 (2012)
Kronz, M.: Some function spaces on spaces of homogeneous type. Manuscr. Math. 106, 219–248 (2001)
Lerner, A.K.: On the John-Strömberg characterization of BMO for nondoubling measures. Real Anal. Exchange 28, 649–660 (2002)
Maasalo, O.E.: Global integrability of p-superharmonic functions on metric spaces. J. Anal. Math. 106, 191–207 (2008)
Mateu, J., Mattila, P., Nicolau, A., Orobitg, J.: BMO for nondoubling measures. Duke Math. J. 102, 533–565 (2000)
Reimann, H.M.: Functions of bounded mean oscillation and quasiconformal mappings. Comment. Math. Helv. 49, 260–276 (1974)
Strömberg, J.-O.: Bounded mean oscillation with Orlicz norms and duality of Hardy spaces. Indiana Univ. Math. J. 28, 511–544 (1979)
J.-O. Strömberg and A. Torchinsky: Weighted Hardy Spaces, Lecture Notes in Mathematics, vol. 1381, Springer, Berlin (1989)
Uchiyama, A.: The construction of certain BMO functions and the corona problem. Pac. J. Math. 99, 183–204 (1982)
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The research is supported by the Academy of Finland.
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Kinnunen, J., Korte, R., Marola, N. et al. A characterization of \({{\mathrm{BMO}}}\) self-maps of a metric measure space. Collect. Math. 66, 405–421 (2015). https://doi.org/10.1007/s13348-014-0126-7
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DOI: https://doi.org/10.1007/s13348-014-0126-7