FRACTAL MEASURES AND MEAN P-VARIATIONS

Recently Strichartz proved that if μ is locally uniformly α-dimensional on Rd, then {A figure is presented}, where 0 ≤ α ≤ d, and BT 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 supT → ∞( 1 Td - α) ∝B|(μf) \ ̂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 S1x = ρ{variant}x, S2x = ρ{variant}x + (1 - ρ{variant}), 1 2 < ρ{variant} < 1 with weights 1 2 each. The index α for μ corresponding to ρ{variant} = (√5 - 1) 2 is calculated. The expression for such α is significantly different from the one obtained by Strichartz. © 1992.