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In this note, we establish a novel maximal inequality of the 2D Youngintegral $\int_a^b\int_c^d FdG$ in terms of the $(p,q)$-bivariation norms ofthe section functions $x\mapsto F(x,y)$ and $y\mapsto F(x,y)$ where$G:[a,b]\times [c,d]\rightarrow \mathbb{R}$ is a controlled path satisfyingfinite $(p,q)$-variation conditions. The proof is reminiscent from the Young'soriginal ideas \cite{young1} in defining two-parameter integrals in terms of$(p,q)$-finite bivariations. Our result complements the standard maximalinequality established by Towghi \cite{towghi1} in terms of joint variations.We apply the maximal inequality to get novel strong approximations for 2D Youngintegrals w.r.t the Brownian local time in terms of number of upcrossings of agiven approximating random walk.

A Maximal Inequality of the 2D Young Integral based on Bivariations