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 L2 inner product
on certain compact supported function spaces.