Riv. Mat. Univ. Parma, Vol. 14, No. 1, 2023

Nicola Arcozzi [a], Alessandro Monguzzi [b] and Maura Salvatori [c]

Ahlfors regular spaces have regular subspaces of any dimension

Pages: 19-32
Received: 28 August 2021
Accepted: 3 January 2022
Mathematics Subject Classification: 28A78, 30L05.
Keywords: Ahlfors-regular metric spaces, imbeddings of metric spaces.
Authors address:
[a]: Alma Mater Studiorum Università di Bologna, Dipartimento di Matematica,Italy.
[b]: Università degli Studi di Bergamo, Dipartimento di Ingegneria Gestionale, dell'Informazione e della Produzione, Italy.
[c]: Università degli Studi di Milano, Dipartimento di Matematica, Italy.

All authors are members of INDAM-GNAMPA. The first and second authors are partially supported by the grant GNAMPA 2020 Alla frontiera tra l'analisi complessa in piĆ¹ variabili e l'analisi armonica. The third author is partially supported by the grant GNAMPA 2020 Fractional Laplacians and subLaplacians on Lie groups and trees. The first author is partially supported by European Unions Horizon 2020 research programme, Marie Sklodowska-Curie grant agreement No. 777822.

Full Text (PDF)

Abstract: We characterize \(Q\)-dimensional Ahlfors regular spaces among trees' boundaries and show how to construct, for each \(0 < \alpha < Q\), an \(\alpha\)-regular subspace. As an application, we give an alternative simple proof of the existence of \(\alpha\)-regular subspaces of a \(Q\)-dimensional complete Ahlfors regular metric space \((X,\rho)\), which was proved in [8].

References
[1]
N. Arcozzi, R. Rochberg, E. T. Sawyer and B. D. Wick, Potential theory on trees, graphs and Ahlfors-regular metric spaces, Potential Anal. 41 (2014), no. 2, 317-366. MR3232028
[2]
M. Christ, A \(T(b)\) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), no. 2, 601-628. MR1096400
[3]
G. David and S. Semmes, Fractured fractals and broken dreams. Self-similar geometry through metric and measure, Oxford Lecture Ser. Math. App., 7, The Clarendon Press, Oxford University Press, New York, 1997. MR1616732
[4]
G. David and S. Semmes, Regular mappings between dimensions, Publ. Mat. 44 (2000), no. 2, 369-417. MR1800814
[5]
K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Math., 85, Cambridge University Press, Cambridge, 1986. MR0867284
[6]
J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), no. 1, 1-61. MR1654771
[7]
J. D. Howroyd, On dimension and on the existence of sets of finite positive Hausdorff measure, Proc. London Math. Soc. 70 (1995), no. 3, 581-604. MR1317515
[8]
E. Järvenpää, M. Järvenpää, A. Käenmäki, T. Rajala, S. Rogovin and V. Suomala, Packing dimension and Ahlfors regularity of porous sets in metric spaces, Math. Z. 266 (2010), no. 1, 83-105. MR2670673
[9]
C. A. Rogers, Hausdorff measures, Cambridge University Press, London-New York, 1970. MR0281862


Home Riv.Mat.Univ.Parma