Davis's inequality for orthogonal martingales under differential subordination (Q1591617)

Let \(X=\{X_t,t\geq 0\}\) and \(Y=\{Y_t,t \geq 0\}\) be two orthogonal martingales such that \(Y\) is differentially subordiated to \(X\), i.e. \([X,Y]_t=0\) for all \(t\geq 0\) and \([X]_t-[Y]_t\) is a nondecreasing and nonnegative function of \(t\), where \(\{[X,Y]_t, t\geq 0\}\) denotes the quadratic covariation process between \(X\) and \(Y\); \([X]= [X,X]\). It is proved that for any \(\lambda\geq 0\), \[ \lambda P\Bigl\{\sup_{t\geq 0} |Y_t|\geq\lambda \Bigr\}\leq {\pi^2\over 8 \beta(2)} \sup_{t\geq 0}E |X_t|, \] where \(\beta(2)\) is the Catalan's constant. The inequality is sharp. Some extensions of this result and the interplay between martingales and harmonic functions are also discussed.