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A note on capacity and Hausdorff measure in homogeneous spaces. (English) Zbl 0873.31013

Let \((X,d,\mu)\) be a space of homogeneous type, that is, \(X\) is a topological space equipped with a quasi-metric \(d\) and a doubling measure \(\mu\). The balls in \(X\) are \(B(x,r)=\{y;d(y,x)<r\}\), the kernels \(k_{\alpha}\), the analogue of the Riesz kernel, is defined by, \[ k_{\alpha}(x,y)=(\mu\overline{B}(x,d(x,y))+ \mu\overline{B}(y,d(x,y)))^{-\alpha} \] The capacity related to this kernel is \(C_{\alpha}\). The Hausdorff measure \(H_{\alpha}\) of order \(\alpha\), \(0<\alpha\leq 1\), is defined by, \(H_{\alpha}(E)=\lim_{\delta\rightarrow 0}H_{\alpha}^{\delta}(E)\), where, \[ H_{\alpha}^{\delta}(E)=\inf\Bigl\{\sum_{i}\mu B(x_{i},r_{i})^{\alpha};E\subset \cup_{i}B(x_{i},r_{i}),\;\mu B(x_{i},r_{i})\leq \delta\Bigr\} \] A homogeneous space of order \(\gamma\), \(0<\gamma <1\), is a space \((X,d',\mu)\) where, \[ |d'(x,z)-d'(y,z)|\leq CR^{1-\gamma}d'(x,y)^{\gamma}, \] for every homogeneous space \((X,d,\mu)\) there is an equivalent quasi-norm \(d'\) satisfying the above condition.
The main result is: If we have a complete homogeneous space of order \(\gamma\), \(0<\gamma<1\), and \(\mu (\{x\})=0\) for every point in \(X\). Let \(0<\alpha <\beta\leq 1\), and \(K\) be a compact set, then, \(C_{\alpha}(K)>0\) implies that \(H_{\alpha}(K)>0\). If in addition the space satisfies the density condition then, \(H_{\beta} (K)>0\) implies that \(C_{\alpha}(K)>0\).

MSC:

31C15 Potentials and capacities on other spaces
43A85 Harmonic analysis on homogeneous spaces
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