The authors construct a modified Green potential in the upper-half space of the
-dimensional Euclidean space. The motivation is the modification of the Poisson kernel by subtracting some special harmonic polynomials which allowed Finkelstein and Scheinberg to solve the Dirichlet problem for the upper half-space with arbitrary continuous data. In the present article, the authors modify the Green function of order
for the upper half-space by subtracting some combinations of suitable ultraspherical (Gegenbauer) polynomials from the kernel. The behavior at infinity of this modified Green function is also given. The finding is that outside a Borel set with controlled capacity, the Green function grows slower than an appropriate power of the absolute value of the argument times an appropriate power of the last (positive) coordinate of the argument. A geometric approach using certain covering with balls of the aforementioned Borel set is also provided.