# Space regularity for evolution operators modeled on H\"ormander vector fields with time dependent measurable coefficients

## TL;DR

This paper establishes regularity and continuity properties of solutions to a heat-type operator on Carnot groups with time-dependent measurable coefficients, extending classical results to a non-smooth coefficient setting.

## Contribution

It proves that solutions gain regularity in space and exhibit time continuity under minimal regularity assumptions on the coefficients, with quantitative Sobolev estimates.

## Key findings

- Solutions are smooth in space if the operator applied to them is smooth.
- Solutions and their derivatives are 1/2-Hölder continuous in time.
- Results hold for both weak and distributional solutions.

## Abstract

We consider a heat-type operator L structured on the left invariant 1-homogeneous vector fields which are generators of a Carnot group, multiplied by a uniformly positive matrix of bounded measurable coefficients depending only on time. We prove that if Lu is smooth with respect to the space variables, the same is true for u, with quantitative regularity estimates in the scale of Sobolev spaces defined by right invariant vector fields. Moreover, the solution and its space derivatives satisfy a 1/2-H\"older continuity estimate with respect to time. The result is proved both for weak solutions and for distributional solutions, in a suitable sense.