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

In this note we prove the following good-$λ$ inequality, for $r>2$, all $λ> 0$, $δ\in \big(0, \frac{1}{2} \big)$ $ν\big\{ V_r(f) > 3 λ; \mathcal{M}(f) \leq δλ\big\} \leq 4 ν\{s(f) > δλ\} + {δ^2 \left(1+\frac{16}{r-2}\right)^2} \cdot ν\big\{ V_r(f) > λ\big\},$ 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.

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