The norm $\left\lVert\cdot\right\rVert_{C^{k,\alpha}_{\delta,0}}$ can refer to any type of tensors depending on the context, such as functions, 1-forms, symmetric 2-tensors, and in some cases can refer to the norm computed in a subregion. Strictly speaking this norm depends on $\kappa$, but we suppress this to avoid cluttering the notation.