FRACTIONAL INTEGRATION FOR IRREGULAR MARTINGALES

We suggest two versions of the Hardy???Littlewood???Sobolev inequality for discrete time martingales. In one version, the fractional integration operator is a martingale transform, however, it may vanish if the filtration is excessively irregular; the second version lacks the martingale property while being analytically meaningful for an arbitrary filtration.

Automated summary

This paper suggests two versions of the Hardy-Littlewood-Sobolev inequality for discrete time martingales. In one version, the fractional integration operator acts as a martingale transform but may vanish if the filtration is excessively irregular. The second version lacks the martingale property but is analytically meaningful for any filtration.