Variational estimates for paraproducts

TL;DR

This paper extends variational norm estimates to paraproducts in both discrete and continuous settings, broadening the scope of previous work by Lepingle, Bourgain, and Pisier-Xu, with a focus on continuous cases and wider exponents.

Contribution

It introduces a generalized family of variational estimates for paraproducts, expanding prior results to include a broader range of exponents and the continuous setting.

Findings

Extended variational estimates to continuous paraproducts.

Broadened the range of variation exponents.

Unified discrete and continuous frameworks.

Abstract

We generalize a family of variation norm estimates of Lepingle with endpoint estimates of Bourgain and Pisier-Xu to a family of variational estimates for paraproducts, both in the discrete and the continuous setting. This expands on work of Friz and Victoir, our focus being on the continuous case and an expanded range of variation exponents.