  A bosonic Laplacian is a conformally invariant second order differential
operator acting on smooth functions defined on domains in Euclidean space and
taking values in higher order irreducible representations of the special
orthogonal group. In this paper, we firstly introduce the motivation for study
of the generalized Maxwell operators and bosonic Laplacians (also known as the
higher spin Laplace operators). Then, with the help of connections between
Rarita-Schwinger type operators and bosonic Laplacians, we solve Poisson's
equation for bosonic Laplacians. A representation formula for bounded solutions
to Poisson's equation in Euclidean space is also provided. In the end, we
provide Green's formulas for bosonic Laplacians in scalar-valued and
Clifford-valued cases, respectively. These formulas reveal that bosonic
Laplacians are self-adjoint with respect to a given $L^2$ inner product on
certain compact supported function spaces.
