Irrigability and Dimension

Abstract

Let μ be a positive Boral measure on ℝ^N which we assume without loss of generality to have total mass 1. If $E^\alpha \left( {\delta _S ,\mu } \right) < \infty$ for some α ∈ [0,1], then we say that μ is irrigable with respect to α. In that case, notice that μ is also β-irrigable for β>α. This observation proves the existence of a critical exponent α associated with μ and defined as the smallest exponent such that μ is α-irrigable. The aim of the chapter is to link this exponent to more classical dimensions associated with μ such as the Hausdorff and Minkowski dimensions of the support of μ. Inequalities between these dimensions and an “irrigability dimension” will be established. A striking result is that when μ is Ahlfors regular, all considered dimensions are equal. To illustrate the results, let us take the case where μ is the Lebesgue measure of a ball in dimension N. We already know in that case that μ is irrigable for every $\alpha > 1 - \tfrac{1} {N}$. What happens if α is critical? Corollary 10.16 gives the answer: if any probability measure μ with a bounded supports is α-irrigable, then α>1/N. Thus the N-dimensional Lebesgue measure is not 1−1/N irrigable. Here the presentation and results follow closely Devillanova’s PhD [28] and Devillanova-Solimini [78].