A self-similar measure on R^n is defined to be a probability measure satisfying
\mu = \sum_{j=1}^N p_j\mu\circ S_j^{-1} + \sum_{j=1}^M q_j(\mu*\mu)\circ T_j^{-1},
where S_j x = \rho_j R_j x + b_j, T_j x = \eta_j Q_j x + c_j are contractive similarities, 0 < \rho_j < 1, 0 < \eta_j < 1/2, 0 < p_j < 1, 0 < q_j < 1; \sum_{j=1}^N p_j + \sum_{j=1}^M q_j = 1, R_j, Q_j are orthogonal matrix and \mu*\mu is the convolution of two measures.
When M = 0, \mu is a linear self-similar measure, we establish the asymptotic behavior of averages of the derivative of
the Fourier transform of \mu, such as
\int_{|x|\le R} |(\frac{\partial}{\partial x})^\alpha \hat\mu(x)|^2 dx = O(R^{n-\beta})
for any order derivation of \hat\mu(x) as R\to\infty under certain additional hypotheses.
When M > 0, \mu is a nonlinear self-similar measure, we get some results of L^p boundedness for maximal operators
of l, from the pointwise asymptotic estimate of the Fourier transform of \mu made by Strichartz.