Littlewood-Paley-Stein functions for Hodge-de Rham and Schr{\"o}dinger operators

TL;DR

This paper investigates Littlewood-Paley-Stein functions linked to Hodge-de Rham and Schr{"o}dinger operators on Riemannian manifolds, establishing boundedness results under Ricci curvature conditions without requiring doubling measures or heat kernel estimates.

Contribution

It provides new boundedness results for Littlewood-Paley-Stein functions on Riemannian manifolds under Ricci curvature conditions, without assuming doubling measures or heat kernel bounds.

Findings

Boundedness of Littlewood-Paley-Stein functions for p in (p1, 2]

Linking Littlewood-Paley-Stein functions to the Riesz Transform

Criteria for boundedness of functions for p > 2

Abstract

We study the Littlewood-Paley-Stein functions associated with Hodge-de Rham and Schr{\"o}dinger operators on Riemannian manifolds. Under conditions on the Ricci curvature we prove their boundedness on L p for p in some interval (p 1 , 2] and make a link to the Riesz Transform. An important fact is that we do not make assumptions of doubling measure or estimates on the heat kernel in this case. For p > 2 we give a criterion to obtain the boundedness of the vertical Littlewood-Paley-Stein function associated with Schr{\"o}dinger operators on L p .