Maximal operators and Fourier transforms of self-similar measures

Abstract

Abstract A self-similar measure on R n is defined to be a probability measure satisfying μ = ∑ j = 1 N p j μ ∘ S j - 1 + ∑ j = 1 M q j ( μ ∗ μ ) ∘ T j - 1 , where Sjx = ρjRjx + bj, Tjx = ηjQjx + cj are contractive similarities, 0 ρ j 1 , 0 η j 1 2 , 0 p j 1 , 0 q j 1 , ∑ j = 1 N p j + ∑ j = 1 M q j = 1 , Rj, Qj are orthogonal matrix and μ ∗ μ is the convolution of two measures. When M = 0, μ is a linear self-similar measure, we establish the asymptotic behavior of averages of the derivative of the Fourier transform of μ, such as ∫ | x | ⩽ R ∂ ∂ x α μ ˆ ( x ) 2 d x = O ( R n - β ) for any order derivation of μ ˆ ( x ) as R → ∞ under certain additional hypotheses. When M > 0, μ is a nonlinear self-similar measure, we get some results of Lp boundedness for maximal operators of μ, from the pointwise asymptotic estimate of the Fourier transform of μ made by Strichartz.