# Growth and asymptotic sets of subharmonic functions (II)

• Volume: 42, Issue: 2, page 449-460
• ISSN: 0214-1493

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## Abstract

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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 &lt; α ≤ n, there exists a subharmonic function u in the Rn+1+ satisfying the growth condition of order α : u(x) ≤ x-αn+1 for 0 &lt; xn+1 &lt; 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.

## How to cite

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Wu, Jang-Mei. "Growth and asymptotic sets of subharmonic functions (II)." Publicacions Matemàtiques 42.2 (1998): 449-460. <http://eudml.org/doc/41344>.

@article{Wu1998,
abstract = {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 &lt; α ≤ n, there exists a subharmonic function u in the Rn+1+ satisfying the growth condition of order α : u(x) ≤ x-αn+1 for 0 &lt; xn+1 &lt; 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.},
author = {Wu, Jang-Mei},
journal = {Publicacions Matemàtiques},
keywords = {Función subarmónica; Comportamiento asintótico; Factores de crecimiento; asymptotic values; subharmonic function; boundary points; Hausdorff dimension},
language = {eng},
number = {2},
pages = {449-460},
title = {Growth and asymptotic sets of subharmonic functions (II)},
url = {http://eudml.org/doc/41344},
volume = {42},
year = {1998},
}

TY - JOUR
AU - Wu, Jang-Mei
TI - Growth and asymptotic sets of subharmonic functions (II)
JO - Publicacions Matemàtiques
PY - 1998
VL - 42
IS - 2
SP - 449
EP - 460
AB - 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 &lt; α ≤ n, there exists a subharmonic function u in the Rn+1+ satisfying the growth condition of order α : u(x) ≤ x-αn+1 for 0 &lt; xn+1 &lt; 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.
LA - eng
KW - Función subarmónica; Comportamiento asintótico; Factores de crecimiento; asymptotic values; subharmonic function; boundary points; Hausdorff dimension
UR - http://eudml.org/doc/41344
ER -

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