# On pointwise exponentially weighted estimates for the Boltzmann equation

## TL;DR

This paper proves that solutions to the non-cutoff homogeneous Boltzmann equation maintain weighted $L^$ bounds over time, assuming they start with such bounds and have propagating weighted $L^1$ bounds, with applications to exponential and Mittag-Leffler weights.

## Contribution

It establishes a general framework linking the propagation of weighted $L^$ bounds to weighted $L^1$ bounds for the Boltzmann equation, including specific weight cases.

## Key findings

- Weighted $L^$ bounds propagate over time under certain conditions.
- The main result applies to exponential and Mittag-Leffler weights.
- Propagation in weighted $L^1$ bounds implies propagation in weighted $L^$ bounds.

## Abstract

In this paper we prove propagation in time of weighted $L^\infty$ bounds for solutions to the non-cutoff homogeneous Boltzmann equation that satisfy propagation in time of weighted $L^1$ bounds. To emphasize that the propagation in time of weighted $L^{\infty}$ bounds relies on the propagation in time of weighted $L^1$ bounds, we express our main result using certain general weights. Consequently we apply the main result to cases of exponential and Mittag-Leffler weights, for which propagation in time of weighted $L^1$ bounds holds.