The maximal function and conditional square function control the variation: An elementary proof

Abstract:

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