We study the relation between the growth of a subharmonic function in the half space Rn+1 + and the size of its asymptotic set. In particular, we prove that for any n ≥ 1 and 0 < α ≤ n, there exists a subharmonic function u in the Rn+1 + satisfying the growth condition of order α : u(x) ≤ x-α n+1 for 0 < xn+1 < 1, such that the Hausdorff dimension of the asymptotic set ∪λ≠0A(λ) is exactly n-α. Here A(λ) is the set of boundary points at which f tends to λ along some curve. This proves the sharpness of a theorem due to Berman, Barth, Rippon, Sons, Fernández, Heinonen, Llorente and Gardiner cumulatively.
@article{urn:eudml:doc:41344,
title = {Growth and asymptotic sets of subharmonic functions (II)},
journal = {Publicacions Matem\`atiques},
volume = {42},
year = {1998},
pages = {449-460},
mrnumber = {MR1677612},
zbl = {0920.31001},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41344}
}
Wu, Jang-Mei. Growth and asymptotic sets of subharmonic functions (II). Publicacions Matemàtiques, Tome 42 (1998) pp. 449-460. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41344/