Classification of solutions to the $Q$-flat and constant $T$-curvature equation on the half-space and ball

For conformal boundary operators associated with the Paneitz operator, we introduce a rigorous definition of the biharmonic Poisson kernel consisting of a pair of kernel functions and derive its explicit representation formula. With this powerful tool, we establish classification theorems of nonnegative solutions to the $Q$-flat and constant $T$-curvature equations on ${\mathbb{R}}_{+}^{n+1}$ and ${\mathbb{B}}^{n+1}$.