Horizontal Growth of Harmonic Functions |
| |
Authors: | Luis Bá ez-Duarte,G. A. Cá mera |
| |
Affiliation: | IVIC-Matemáticas, Venezuela |
| |
Abstract: | We show that positive harmonic functions in the upper halfplane grow at most quadratically in horizontal bands. This bound is sharp in a sense to be specified, which, at least implies that there are examples growing as fast as any power under 2. These results are extended to positive harmonic functions in a half-space of R n +1, with points represented by ( x , y ), where x ∈R n , and y ∈R, the sharp maximum rate of growth being now ¦ x ¦ n +1. The case of Poisson integrals of functions in Lp ( dx /(1+(¦ x ¦)2 )( n +1)/2) is also taken up; the bound condition is then O (¦ x ¦( n +1)/ p ). |
| |
Keywords: | |
|
|