  The regularity theory of the Campanato space
$\mathcal{L}^{(q,\lambda)}_k(\Omega)$ has found many applications within the
regularity theory of solutions to various geometric variational problems. Here
we extend this theory from single-valued functions to multi-valued functions,
adapting for the most part Campanato's original ideas (\cite{campanato}). We
also give an application of this theory within the regularity theory of
stationary integral varifolds. More precisely, we prove a regularity theorem
for certain \textit{blow-up classes} of multi-valued functions, which typically
arise when studying blow-ups of sequences of stationary integral varifolds
converging to higher multiplicity planes or unions of half-planes. In such a
setting, based in part on ideas in \cite{wickstable}, \cite{minterwick}, and
\cite{beckerwick}, we are able to deduce a boundary regularity theory for
multi-valued harmonic functions; such a boundary regularity result would appear
to be the first of its kind for the multi-valued setting. In conjunction with
\cite{minter}, the results presented here establish a regularity theorem for
stable codimension one stationary integral varifolds near classical cones of
density $\frac{5}{2}$.
