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ON THE DOUBLING CONDITION IN THE INFINITE-DIMENSIONAL SETTING

Part of: Spaces with richer structures Abstract harmonic analysis

Published online by Cambridge University Press:  09 February 2023

DARIUSZ KOSZ Open the ORCID record for DARIUSZ KOSZ [Opens in a new window]
DARIUSZ KOSZ*
Affiliation:
Basque Center for Applied Mathematics, 48009 Bilbao, Spain and Wrocław University of Science and Technology, 50-370 Wrocław, Poland
*
e-mail: dkosz@bcamath.org
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Abstract

We present a systematic approach to the problem whether a topologically infinite-dimensional space can be made homogeneous in the Coifman–Weiss sense. The answer to the question is negative, as expected. Our leading representative of spaces with this property is $\mathbb {T}^\omega = \mathbb {T} \times \mathbb {T} \times \cdots $ with the natural product topology.

Keywords

doubling conditionquasimetric spaceinfinite-dimensional torus

MSC classification

Primary: 43A70: Analysis on specific locally compact and other abelian groups
Secondary: 54E99: None of the above, but in this section

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 108 , Issue 3 , December 2023 , pp. 480 - 491
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The author was supported by the Basque Government (BERC 2022-2025), by the Spanish State Research Agency (SEV-2017-0718) and by the Foundation for Polish Science (START 032.2022).

References

Aimar, H., Iaffei, B. and Nitti, L., ‘On the Macías–Segovia metrization of quasi-metric space’, Rev. Un. Mat. Argentina 41 (1998), 67–75.Google Scholar
Bendikov, A. D., Potential Theory on Infinite-dimensional Abelian Groups, de Gruyter Studies in Mathematics, 21 (de Gruyter, Berlin, 1995).10.1515/9783110876840CrossRefGoogle Scholar
Coifman, R. R. and Weiss, G., Analyse Harmonique Non-commutative sur Certaines Espaces Homogènes, Lecture Notes in Mathematics, 242 (Springer-Verlag, Berlin, 1971).10.1007/BFb0058946CrossRefGoogle Scholar
Edgar, G., Integral, Probability, and Fractal Measures, Undergraduate Texts in Mathematics (Springer-Verlag, New York, 1998).10.1007/978-1-4757-2958-0CrossRefGoogle Scholar
Engelking, R., Dimension Theory, North-Holland Mathematical Library, 19 (North-Holland Publishing Company, Amsterdam–New York, 1978).Google Scholar
Fernández, E., Análisis de Fourier en el Toro Infinito-dimensional, PhD Thesis, Universidad de La Rioja, 2019.Google Scholar
Fernández, E. and Roncal, L., ‘On the absolute divergence of Fourier series on the infinite-dimensional torus’, Colloq. Math. 157 (2019), 143–155.10.4064/cm7568-6-2018CrossRefGoogle Scholar
Fernández, E. and Roncal, L., ‘A decomposition of Calderón–Zygmund type and some observations on differentiation of integrals on the infinite-dimensional torus’, Potential Anal. 53 (2020), 1449–1465.10.1007/s11118-019-09813-8CrossRefGoogle Scholar
Hytönen, T., ‘A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa’, Publ. Mat. 54 (2010), 485–504.10.5565/PUBLMAT_54210_10CrossRefGoogle Scholar
Kosz, D., ‘On differentiation of integrals in the infinite-dimensional torus’, Studia Math. 258 (2021), 103–119.10.4064/sm191001-10-2CrossRefGoogle Scholar
Kosz, D., Martínez-Perales, J., Paternostro, V., Rela, E. and Roncal, L., ‘Maximal operators on the infinite-dimensional torus’, Math. Ann., to appear.Google Scholar
Luukkainen, J. and Saksman, E., ‘Every complete doubling metric space carries a doubling measure’, Proc. Amer. Math. Soc. 126 (1998), 531–534.10.1090/S0002-9939-98-04201-4CrossRefGoogle Scholar
Paluszyński, M. and Stempak, K., ‘On quasi-metric and metric spaces’, Proc. Amer. Math. Soc. 137 (2009), 4307–4312.10.1090/S0002-9939-09-10058-8CrossRefGoogle Scholar
Stempak, K., ‘On some structural properties of spaces of homogeneous type’, Taiwanese J. Math. 19 (2015), 603–613.10.11650/tjm.19.2015.3428CrossRefGoogle Scholar
Vol’berg, A. L. and Konyagin, S. V., ‘On measures with the doubling condition’, Math. USSR Izv. 30 (1988), 629–638.10.1070/IM1988v030n03ABEH001034CrossRefGoogle Scholar
Wei, C., Wen, S. and Wen, Z., ‘Doubling measures on uniform Cantor sets’, J. Math. Anal. Appl. 430 (2015), 500–516.10.1016/j.jmaa.2015.05.002CrossRefGoogle Scholar