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

Abstract

A metric measure space (X, d, μ) is said to satisfy the strong annular decay condition if there is a constant C > 0 such that μ(B(x,R) \ B(x, r)) ≤ C R – r R μ(B(x,R)) for each x ∈ X and all 0 < r ≤ R. If d∞ is the distance induced by the ∞-norm in RN, we construct examples of singular measures μ on RN such that (RN, d∞, μ) satisfies the strong annular decay condition.