The Maximal Function and Square Function Control the Variation: An Elementary Proof

In this note we prove the following good-$\lambda$ inequality, for $r>2$, all $\lambda > 0$, $\delta \in \big(0, \frac{1}{2} \big)$ $$\nu\big\{ V_r(f) > 3 \lambda ; \mathcal{M}(f) \leq \delta \lambda\big\} \leq 4 \nu\{s(f) > \delta \lambda\} + {\delta^2 \left(1+\frac{16}{r-2}\right)^2} \cdot \nu\big\{ V_r(f) > \lambda\big\},$$ where $\mathcal{M}(f)$ is the martingale maximal function, $s(f)$ is the conditional martingale square function. This immediately proves that $V_r(f)$ is bounded on $L^p$, $1 < p <\infty$ and moreover is integrable when the maximal function is.