On a class of singular measures satisfying a strong annular decay condition

Abstract

A metric measure space ( X , d , μ ) (X, d, \mu ) is said to satisfy the strong annular decay condition if there is a constant C > 0 C>0 such that μ ( B ( x , R ) ∖ B ( x , r ) ) ≤ C R − r R μ ( B ( x , R ) ) \begin{equation*} \mu \big ( B(x, R) \setminus B(x,r) \big ) \leq C\, \frac {R-r}{R}\, \mu (B(x,R)) \end{equation*} for each x ∈ X x\in X and all 0 > r ≤ R 0>r \leq R . If d ∞ d_{\infty } is the distance induced by the ∞ \infty -norm in R N \mathbb {R}^N , we construct examples of singular measures μ \mu on R N \mathbb {R}^N such that ( R N , d ∞ , μ ) (\mathbb {R}^N, d_{\infty }, \mu ) satisfies the strong annular decay condition.