# A generalized Cauchy-Lipschitz theorem in low regularity spaces

## TL;DR

This paper establishes well-posedness for a class of differential equations in low regularity spaces, extending classical results to include integro-differential operators with controlled derivative loss.

## Contribution

It generalizes the Cauchy-Lipschitz theorem to low regularity spaces and includes integro-differential operators with limited derivative loss.

## Key findings

- Proves well-posedness for abstract differential equations with low regularity.
- Extends classical Lipschitz-based results to more general operators.
- Identifies conditions on derivative loss for operator dominance.

## Abstract

We prove well-posedness for some abstract differential equations of the first order. Our result covers the usual case of Lipschitz composition operators. It also contains the case of some integro-differential operators acting on spaces with low regularity indexes. The loss of derivatives induced by such operators has to be lower than one, in order to be dominated by the first order derivative involved in the problem.