On harmonic function spaces, we define shift operators
using zonal harmonics and partial derivatives, and develop
their basic properties.
These operators turn out to be multiplications by the
coordinate variables followed by projections on harmonic
subspaces.
We introduce large families of reproducing kernel Hilbert
spaces of harmonic functions on the unit ball of
$\mathbb R^n$ and investigate the action of the shift
operators on them.
We prove a dilation result for a commuting row contraction
which is also what we call of Laplacian type.
As a consequence, we show that the norm of one of our
spaces $\breve{\mathcal G}$ is maximal among those spaces
with contractive norms on harmonic polynomials.
We then obtain a von Neumann inequality for harmonic
polynomials of a commuting Laplacian-type row contraction.
This yields the maximality of the operator norm of a
harmonic polynomial of the shift on $\breve{\mathcal G}$
making this space a natural harmonic counterpart of the
Drury-Arveson space.