A note on the Taylor estimates of iterated paraproducts

TL;DR

This paper extends Taylor estimates for iterated paraproducts in paracontrolled calculus to higher regularities, improving understanding of their local behavior beyond previous Fourier-based bounds.

Contribution

It generalizes existing pointwise estimates for paraproducts and their iterates to cases with higher regularities, enhancing analytical tools in paracontrolled calculus.

Findings

Extended Taylor estimates to higher regularities

Established local bounds for iterated paraproducts

Improved analytical understanding of paraproduct behavior

Abstract

Bony's paraproduct is one of the main tools in the theory of paracontrolled calculus. The paraproduct is usually defined via Fourier analysis, so it is not a local operator. In the previous researches [7, 8], however, the author proved that the pointwise estimate like (1.2) holds for the paraproduct and its iterated versions when the sum of the regularities is smaller than 1. The aim of this article is to extend these results for higher regularities.