Horizontal Growth of Harmonic Functions

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 L_p ( dx /(1+(¦ x ¦)² )^{( n +1)/2}) is also taken up; the bound condition is then O (¦ x ¦^{( n +1)/ p} ).