# The derivatives of the heat kernel on symmetric spaces

## TL;DR

This paper derives estimates for the derivatives of the heat kernel on symmetric spaces and uses these to analyze the boundedness of related operators on locally symmetric spaces.

## Contribution

It provides new derivative estimates of the heat kernel on symmetric spaces and applies them to study operator boundedness in a broad class of locally symmetric spaces.

## Key findings

- Established derivative estimates for the heat kernel on symmetric spaces
- Proved $L^{p}$-boundedness of Littlewood-Paley-Stein operators
- Analyzed the Laplacian of the heat operator on locally symmetric spaces

## Abstract

We derive estimates of the derivatives of the heat kernel on noncompact symmetric spaces and on locally symmetric spaces. Applying these estimates we study the $L^{p}$-boundedness of Littlewood-Paley-Stein operators and the Laplacian of the heat operator on a wide class of locally symmetric spaces.