The maximal function and conditional square function control the variation: an elementary proof (Q2809214)

Let \((X,{\mathcal F},\nu)\) be a \(\sigma\)-finite measure space and let \(({\mathcal F_n}; n \in \mathbb{Z})\) be a sequence of \(\sigma\)-fields such that \({\mathcal F}_n\subseteq {\mathcal F}_{n+1}\subseteq {\mathcal F}\). For a martingale \((f_n: n\in\mathbb Z)\), consider the maximal and the conditional martingale square function defined, respectively, by NEWLINE\[NEWLINE M(f)= \sup_{n\in \mathbb{Z}} |f_n|, \;\;s(f)=\left( \sum_{n\in \mathbb{Z}} \mathbb{E}[|f_n-f_{n-1}|^2 |{\mathcal F}_{n-1}]\right)^{1/2}.NEWLINE\]NEWLINENEWLINENEWLINEAlso, define for \(r\geq 1\) the following family of \(r\)-variation operators controlling the oscillation NEWLINE\[NEWLINEV_r(f)=\sup_{n_0<n_1<\cdots <n_j} \left( \sum_{j=1}^J |f_{n_j}-f_{n_{j-1}} |^r \right)^{1/r}.NEWLINE\]NEWLINE It is a result of \textit{D. Lepingle} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 36, 295--316 (1976; Zbl 0325.60047)] that \(V_r\) is bounded on \(L^p(\nu)\) for \(p>1\) and satisfies a weak-type boundedness for \(p=1\).NEWLINENEWLINEThe main result of the paper is the proof of the following good-\(\lambda\) inequality, for \(r>2\) and all \(\lambda>0\), \(\delta \in (0,1/2)\): NEWLINE\[NEWLINE\nu\{ V_r(f) >3\lambda; Mf< \delta\lambda\} \lesssim \nu\{s(f)> \delta \lambda\} +\frac{\delta^2}{(r-2)^2} \nu\{V_r(f)>\lambda\}.NEWLINE\]NEWLINE This result proves that, for all \(p\in (0,+\infty)\) and \(r>2\), \(V_r(f)\in L^p(X,\nu)\) whenever \(s(f)\) and \(M(f)\) are in \(L^p(X,\nu)\). In this way, a new proof of Lépingle's result is provided by means of establishing a relationship between the maximal function, the conditional square function and the variation operator.