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Ahlfors \(Q\)-regular spaces with arbitrary \(Q>1\) admitting weak Poincaré inequality. (English) Zbl 0962.30006

Geom. Funct. Anal. 10, No. 1, 111-123 (2000); erratum ibid. 12, 650 (2002).
A metric space \(X\) with a Borel measure \(\mu\) is Ahlfors \(Q\)-regular if \(\tau^Q/C\leq \mu(B(x,r))\leq Cr^Q\) for all \(x\in X\) and \(0< r\leq \text{diam}(X)\). The space \(X\) admits a weak \((1,1)\)-Poincaré inequality if for all balls \(B\), \[ \int_B|u- u_B|d\mu\leq C(\text{diam }B)\Biggl(\int_{\sigma B}\rho d\mu\Biggr) \] for bounded continuous functions in the enlarged ball \(\sigma B\) and \(\rho\) is an upper gradient of \(u\), see [J. Cheeger, Geom. Funct. Anal. 9, No. 3, 428-517 (1999; Zbl 0942.58018)]. For some values of \(Q\) M. Bourdon and H. Pajot [Proc. Am. Math. Soc. 127, No. 8, 2315-2324 (1999; Zbl 0924.30030)] constructed Ahlfors \(Q\)-regular spaces satisfying a weak \((1,1)\)-Poincaré inequality. The author gives an ingenious construction of these spaces for all \(Q>1\). The construction is based on curve family methods where certain points of \([0,1]\times K\), \(K\) a Cantor set, are identified.

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
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