Maximal operators and Fourier transforms of self-similar measures

A self-similar measure on is defined to be a probability measure satisfying where Sjx = ρjRjx + bj, Tjx = ηjQjx + cj are contractive similarities, 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 for any order derivation of 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.