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

Kevin Hughes, Ben Krause, Bartosz Trojan

In this note we prove the following good-$\lambda$ inequality, for $r>2$, all $\lambda >0$, $\delta \in (0,\frac{1}{2})$ \[ ν\big\{ V_r(f) > 3 λ; \mathcal{M}(f) \leq δλ\big\} \leq 4 ν\{s(f…