Maximal operators and Fourier transforms of self-similar measures

Yong Lin Department of Mathematics, School of Information, Renmin University of China, Beijing 100872, China Huo-Jun Ruan Department of Mathematics, Zhejiang University, Hangzhou 310027, China

TBD mathscidoc:2207.43005

Chaos, Solitons & Fractals, 27, (1), 121-126, 2006.1
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.
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@inproceedings{yong2006maximal,
  title={Maximal operators and Fourier transforms of self-similar measures},
  author={Yong Lin, and Huo-Jun Ruan},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220707112146621628550},
  booktitle={Chaos, Solitons & Fractals},
  volume={27},
  number={1},
  pages={121-126},
  year={2006},
}
Yong Lin, and Huo-Jun Ruan. Maximal operators and Fourier transforms of self-similar measures. 2006. Vol. 27. In Chaos, Solitons & Fractals. pp.121-126. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220707112146621628550.
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