Fractal measures and mean p-variations

Abstract

Recently Strichartz proved that if μ is locally uniformly α-dimensional on R d , then , where 0 ⩽ α ⩽ d , and B T denotes the ball of radius T center at 0; if μ is self-similar and satisfies a certain open set condition, he also obtained a formula for the α so that 0 < lim sup T → ∞ ( 1 T d − α ) ∝ B T ¦( μ f ) \ ̂ s| 2 < ∞ . The α can serve, in some sense, as the dimensional index of the measure μ. By using the mean p -variation and the Tauberian theorems, we extend the first inequality and its variants to p , q forms, and give necessary and sufficient conditions on μ for such inequalities to hold; we then use the mean quadratic variation to study some self-similar measures μ on R which do not satisfy the open set condition: the μ's that are constructed from S 1 x = ϱx, S 2 x = ϱx + (1 − ϱ), 1 2 < ϱ < 1 with weights 1 2 each. The index α for μ corresponding to ϱ = (√5 − 1) 2 is calculated. The expression for such α is significantly different from the one obtained by Strichartz.