## 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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