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A note on higher dimensional $p$-variation

We discuss $p$-variation regularity of real-valued functions defined on $[0,T]^2$, based on rectangular increments. When $p>1$, there are two slightly different notions of $p$-variation; both of which are useful in the context of Gaussian rough paths. Unfortunately, these concepts were blurred in previous works; the purpose of this note is to show that the aforementioned notions of $p$-variations are "$ε$-close". In particular, all arguments relevant for Gaussian rough paths go through with minor notational changes.