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 thatfor each x ∈ X and all 0 < r ≤ R. If d∞ is the distance induced by the ∞-norm in R N , we construct examples of singular measures µ on R N such that (R N , d∞, µ) satisfies the strong annular decay condition.for each x ∈ X and all 0 < r ≤ R. Whenever the ambient metric space is fixed we will often say that the measure µ itself satisfies a δ-ADC. The case δ = 1 is special in some senses. The 1-ADC is often known as the strong annular decay condition in the literature. Observe that the 1-ADC implies the δ-ADC for any δ ∈ (0, 1].The δ-ADC is closely connected to the doubling property. We say that a positive Borel measure µ on a metric space (for each x ∈ X and any R > 0.It is easy to see that a measure satisfying a δ-ADC for some δ ∈ (0, 1] is doubling (with a doubling constant depending on δ and the constant in the δ-ADC). Conversely, if (X, d) is geodesic then any doubling measure on X satisfies a δ-ADC for some δ ∈ (0, 1) only depending on the doubling constant ([7], see also [2] for an elementary proof when X = R N ). Thus in a geodesic metric space, the doubling condition is equivalent to the δ-ADC for some δ ∈ (0, 1).

Quantitative ergodic theorems for actions of groups of polynomial growth

“…A slight variant on manifold was introduced by Colding and Minicozzi [15], which they called the ǫ-volume regularity property. In recent years the ǫ-annular decay property has been widely exploited in harmonic analysis, see [3,4,35,36,42,59] for more details.…”