# The sharp Gagliardo--Nirenberg--Sobolev inequality in quantitative form

## TL;DR

This paper proves stability estimates for the sharp Gagliardo--Nirenberg--Sobolev inequalities using a dimension reduction approach and recent weighted Sobolev inequality stability results.

## Contribution

It introduces new quantitative stability estimates for the sharp Gagliardo--Nirenberg--Sobolev inequalities based on recent advances in weighted Sobolev inequalities.

## Key findings

- Established stability estimates for the inequalities
- Utilized dimension reduction and recent weighted Sobolev stability results
- Enhanced understanding of the sharp inequality's stability properties

## Abstract

Using a dimension reduction argument and a stability version of the weighted Sobolev inequality on half space recently proved by Seuffert, we establish, in this paper, some stability estimates (or quantitative estimates) for a family of the sharp Gagliardo--Nirenberg--Sobolev inequalities due to Del Pino and Dolbeault \cite{DelPino}.