# Weak $(1,1)$ Boundedness of Riesz Transforms on Vector Bundles

## TL;DR

This paper proves the weak (1,1) boundedness of Riesz transforms associated with Schrödinger operators on vector bundles under certain geometric and heat kernel conditions, advancing understanding of their analytical properties.

## Contribution

It establishes weak (1,1) boundedness for a broad class of Riesz transforms on vector bundles, using generalized volume doubling and heat kernel estimates.

## Key findings

- Weak (1,1) boundedness proved for Riesz transforms on vector bundles.
- Results apply under Gaussian or sub-Gaussian heat kernel bounds.
- Analysis relies on derivative estimates of Bismut type.

## Abstract

The weak $(1,1)$ boundedness of (local) Riesz transforms corresponding to a large class of Schr\"{o}dinger operators on vector bundles is proved, mainly assuming the generalized volume doubling condition, either Gaussian or sub-Gaussian upper bounds for the heat kernel only in short time, and derivative estimates of Bismut type for the corresponding semigroup.