Some Asymptotic Properties of the Convolution Transforms of Fractal Measures

We study the asymptotic behavior near the boundary of u(x, y) = Ky * μ (x), defined on the half-space +×N by the convolution of an approximate identity Ky(·) (y > 0) and a measure μ on N. 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.