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BMO Spaces on Weighted Homogeneous Trees

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Abstract

We consider an infinite homogeneous tree \({\mathcal {V}}\) endowed with the usual metric d defined on graphs and a weighted measure \(\mu \). The metric measure space \(({\mathcal {V}},d,\mu )\) is nondoubling and of exponential growth, hence the classical theory of Hardy and BMO spaces does not apply in this setting. We introduce a space \(BMO(\mu )\) on \(({\mathcal {V}},d,\mu )\) and investigate some of its properties. We prove in particular that \(BMO(\mu )\) can be identified with the dual of a Hardy space \(H^1(\mu )\) introduced in a previous work and we investigate the sharp maximal function related with \(BMO(\mu )\).

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References

  1. Arditti, L., Tabacco, A., Vallarino, M.: Hardy spaces on weighted homogeneous trees. In: Boggiatto, P., et al. (eds.) Advances in Microlocal and Time-Frequency Analysis. Applied and Numerical Harmonic Analysis, pp. 21–39. Birkhäuser, Cham (2020)

    Chapter  Google Scholar 

  2. Bourbaki, N.: Topological Vector Spaces, Chapters 1–5, Elements of Mathematics (Berlin). Springer, Berlin (1987)

    Book  Google Scholar 

  3. Carbonaro, A., Mauceri, G., Meda, S.: \(H^1\) and \(BMO\) for certain locally doubling metric measure spaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8(3), 543–582 (2009)

    MathSciNet  MATH  Google Scholar 

  4. Celotto, D., Meda, S.: On the analogue of the Fefferman–Stein theorem on graphs with the Cheeger property. Ann. Mat. Pura Appl. (4) 197(5), 1637–1677 (2018)

    Article  MathSciNet  Google Scholar 

  5. Coifman, R.R., Weiss, G.: Analyse Harmonique Non Commutative Sur Certains Espaces Homogenes. Lecture Notes in Mathematics, vol. 242. Springer, New York (1971)

  6. Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83(4), 569–645 (1977)

    Article  MathSciNet  Google Scholar 

  7. Fefferman, C., Stein, E.M.: \(H^p\) spaces of several variables. Acta Math. 129, 137–193 (1972)

    Article  MathSciNet  Google Scholar 

  8. Folland, G.B., Stein, E.M.: Hardy Spaces on Homogeneous Groups. Princeton University Press, Princeton (1982)

    MATH  Google Scholar 

  9. Feneuil, J.: Hardy and BMO spaces on graphs, application to Riesz transform. Potential Anal. 45(1), 1–54 (2016)

    Article  MathSciNet  Google Scholar 

  10. Hebisch, W., Steger, T.: Multipliers and singular integrals on exponential growth groups. Math. Z. 245(1), 37–61 (2003)

    Article  MathSciNet  Google Scholar 

  11. John, F., Nirenberg, L.: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14, 415–426 (1961)

    Article  MathSciNet  Google Scholar 

  12. Ionescu, A.D.: Fourier integral operators on noncompact symmetric spaces of real rank one. J. Funct. Anal. 174(2), 274–300 (2000)

    Article  MathSciNet  Google Scholar 

  13. Martini, A., Ottazzi, A., Vallarino, M.: Spectral multipliers for sub-Laplacians on solvable extensions of stratified groups. J. Anal. Math. 136, 357–397 (2018)

    Article  MathSciNet  Google Scholar 

  14. Mauceri, G., Meda, S., Vallarino, M.: Harmonic Bergman spaces, the Poisson equation and the dual of Hardy-type spaces on certain noncompact manifolds. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 14(4), 1157–1188 (2015)

    MathSciNet  MATH  Google Scholar 

  15. Stein, E.M.: Harmonic Analysis. Princeton University Press, Princeton (1993)

    Google Scholar 

  16. Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Mathematical Series 32. Princeton University Press, Princeton, N.J. (1971)

    MATH  Google Scholar 

  17. Taylor, M.: Hardy spaces and BMO on manifolds with bounded geometry. J. Geom. Anal. 19(1), 137–190 (2009)

    Article  MathSciNet  Google Scholar 

  18. Vallarino, M.: Spaces \(H^1\) and \(BMO\) on \(ax+b\)-groups. Collect. Math. 60(3), 277–295 (2009)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

Work partially supported by the MIUR project “Dipartimenti di Eccellenza 2018-2022” (CUP E11G18000350001). The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

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Authors and Affiliations

  1. Dipartimento di Scienze Matematiche “Giuseppe Luigi Lagrange”, Dipartimento di Eccellenza 2018-2022, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Turin, Italy

    Laura Arditti, Anita Tabacco & Maria Vallarino

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  1. Laura Arditti
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  2. Anita Tabacco
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  3. Maria Vallarino
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Correspondence to Anita Tabacco.

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Dedicated to Guido Weiss on the occasion of his 90th birthday.

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Arditti, L., Tabacco, A. & Vallarino, M. BMO Spaces on Weighted Homogeneous Trees. J Geom Anal 31, 8832–8849 (2021). https://doi.org/10.1007/s12220-020-00435-w

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