Logarithmic estimates for submartingales and their differential subordinates (Q639329)

The author establishes several sharp inequalities for (sub)martin\-gale sequences \(f=(f_n)_{n\geq 0}\) and sequences \(g=(g_n)_{n\geq 0}\) that are \(\alpha\)-subordinate (or differentially subordinate) to \(f\). The first main result states that, if \(g\) is \(\alpha\)-subordinate to a nonnegative submartingale \(f\), \(\alpha\in [0,1]\), then, for any \(K>1\), \[ \|g\|_1\leq K \sup_n \operatorname{E}f_n\log^+ f_n +L(K,\alpha), \] with an explicitly determined constant \(L(K,\alpha)\) which is best possible. Furthermore, if \(K\leq 1\), the inequality does not hold in general with any universal \(L(K,\alpha)<\infty\). Related sharp logarithmic inequalities are derived for \(g\) and its one-sided maximal function in the case when both \(f\) and \(g\) are martingales and \(g\) is differentially subordinate to \(f\). The results are proved in a more general continuous-time settings for \(X\) and \(Y\) real-valued càdlàg semimartingales. As an application, the author derives logarithmic estimates for smooth functions on Euclidean domains.