数学物理学报(英文版) ›› 2012, Vol. 32 ›› Issue (6): 2096-2104.doi: 10.1016/S0252-9602(12)60162-3

• 论文 • 上一篇    下一篇

SOME ASYMPTOTIC PROPERTIES OF THE CONVOLUTION TRANSFORMS OF FRACTAL MEASURES

曹丽   

  1. School of Mathematics, Hefei University of Technology, Hefei 230009, China;
    School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
  • 收稿日期:2011-08-02 出版日期:2012-11-20 发布日期:2012-11-20
  • 基金资助:

    Research supported by the National Natural Science Foundation of China (10671150).

SOME ASYMPTOTIC PROPERTIES OF THE CONVOLUTION TRANSFORMS OF FRACTAL MEASURES

 CAO Li   

  1. School of Mathematics, Hefei University of Technology, Hefei 230009, China;
    School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
  • Received:2011-08-02 Online:2012-11-20 Published:2012-11-20
  • Supported by:

    Research supported by the National Natural Science Foundation of China (10671150).

摘要:

We study the asymptotic behavior near the boundary of u(x, y) = Ky * μ (x), defined on the half-space R+×RN by the convolution of an approximate identity Ky(·) (y >0) and a measure μ on RN. The Poisson and the heat kernel are unified as special cases in our setting. We are mainly interested in the relationship between the rate of growth at boundary of u and the s-density of a singular measure μ. Then a boundary limit theorem of Fatou´s type for singular measures is proved. Meanwhile, the asymptotic behavior of a quotient of Kμ and Kν is also studied, then the corresponding Fatou-Doob´s boundary relative limit is obtained. In particular, some results about the singular boundary behavior of harmonic and heat functions can be deduced simultaneously from ours. At the end, an application in fractal geometry is given.

关键词: fractal density, harmonic function, convolution, singular measure

Abstract:

We study the asymptotic behavior near the boundary of u(x, y) = Ky * μ (x), defined on the half-space R+×RN by the convolution of an approximate identity Ky(·) (y >0) and a measure μ on RN. The Poisson and the heat kernel are unified as special cases in our setting. We are mainly interested in the relationship between the rate of growth at boundary of u and the s-density of a singular measure μ. Then a boundary limit theorem of Fatou´s type for singular measures is proved. Meanwhile, the asymptotic behavior of a quotient of Kμ and Kν is also studied, then the corresponding Fatou-Doob´s boundary relative limit is obtained. In particular, some results about the singular boundary behavior of harmonic and heat functions can be deduced simultaneously from ours. At the end, an application in fractal geometry is given.

Key words: fractal density, harmonic function, convolution, singular measure

中图分类号: 

  • 31B25