Growth estimates and Phragmen-Lindelof principles for half space problems

Pointwise growth estimates for the n-dimensional {dollar}(nge2){dollar} half space Dirichlet and Neumann Poisson integrals are given. They are shown to be the best estimates possible within the class of functions for which the integrals converge. A modified Poisson kernel can be formed by subtracting M terms from the Fourier expansion (in Gegenbauer polynomials) of the Poisson kernel. With Dirichlet data, the resulting modified Poisson integral satisfies {dollar}u(x)-o(vert xvertsp{lcub}M=1{rcub}{lcub}rm sec{rcub}sp{lcub}n-1{rcub}theta)(xtoinfty,xsb{lcub}n{rcub}>0){dollar} where {dollar}theta{dollar} is the angle between x and the normal to the half space {dollar}xsb{lcub}n{rcub}=0.{dollar} Here the data is {dollar}f:IRsp{lcub}n-1{rcub}toIR{dollar} and satisfies {dollar}intsb{lcub}IRsp{lcub}n-1{rcub}{rcub}vert f(y)vert(vert yvertsp{lcub}M+n{rcub}+1)sp{lcub}-1{rcub}dt0{dollar} a similar type of modified kernel is used to give the asymptotic expansion of the Poisson integral as {dollar}vert xverttoinfty.{dollar} Using the Henstock-Kurzweil integral, growth estimates for conditionally convergent Poisson integrals are also given.; A Phragmen-Lindelof principle that takes into account the above angular blow-up is proved. This is done by means of barriers on cusped sub-domains of the half space. This gives an extension of the Phragmen-Lindelof principles of Wolf (1939) and Yoshida (1981) and leads to a uniqueness theorem. Uniqueness is also proven directly using a spherical harmonics expansion.