A Fubini type theorem for rough integration

We develop the integration theory of two-parameter controlled paths $Y$ allowing us to define integrals of the form \begin{equation} \int_{[s,t] \times [u,v]} Y_{r,r'} \;d(X_{r}, X_{r'}) \end{equation} where $X$ is the geometric $p$-rough path that controls $Y$. This extends to arbitrary regularity the definition presented for $2\leq p<3$ in the recent paper of Hairer and Gerasimovi\v{c}s where it is used in the proof of a version of H\"{o}rmander's theorem for a class of SPDEs. We extend the Fubini type theorem of the same paper by showing that this two-parameter integral coincides with the two iterated one-parameter integrals \[ \int_{[s,t] \times [u,v]} Y_{r,r'} \;d(X_{r}, X_{r'}) = \int_{s}^{t} \int_{u}^{v} Y_{r,r'} \;dX_{r'} \;dX_{r'} = \int_{u}^{v} \int_{s}^{t} Y_{r,r'} \;dX_{r} \;dX_{r'}. \] A priori these three integrals have distinct definitions, and so this parallels the classical Fubini's theorem for product measures. By extending the two-parameter Young-Towghi inequality in this context, we derive a maximal inequality for the discrete integrals approximating the two-parameter integral. We also extend the analysis to consider integrals of the form \begin{equation*} \int_{[s,t] \times [u,v]} Y_{r,r'} \; d(X_{r}, \tilde{X}_{r'}) \end{equation*} for possibly different rough paths $X$ and $\tilde{X}$, and obtain the corresponding Fubini type theorem. We prove continuity estimates for these integrals in the appropriate rough path topologies. As an application we consider the signature kernel, which has recently emerged as a useful tool in data science, as an example of a two-parameter controlled rough path which also solves a two-parameter rough integral equation.