  We prove an interior Schauder estimate for the Laplacian on metric products
of two dimensional cones with a Euclidean factor, generalizing the work of
Donaldson and reproving the Schauder estimate of Guo-Song. We characterize the
space of homogeneous subquadratic harmonic functions on products of cones, and
identify scales at which geodesic balls can be well approximated by balls
centered at the apex of an appropriate model cone. We then locally approximate
solutions by subquadratic harmonic functions at these scales to measure the
H\"older continuity of second derivatives.
