# Gagliardo-Nirenberg Inequality for rearrangement-invariant Banach function spaces

## TL;DR

This paper extends the classical Gagliardo-Nirenberg inequality to rearrangement-invariant Banach function spaces, including Orlicz and Lorentz spaces, establishing optimality using scaling arguments.

## Contribution

It introduces a generalized Gagliardo-Nirenberg inequality for rearrangement-invariant spaces and proves its optimality, broadening the scope beyond classical Lebesgue spaces.

## Key findings

- Extended Gagliardo-Nirenberg inequality to Orlicz and Lorentz spaces
- Proved the inequality's optimality using scaling arguments
- Provided corollaries for intermediate derivatives in these spaces

## Abstract

The classical Gagliardo-Nirenberg interpolation inequality is a well-known estimate which gives, in particular, an estimate for the Lebesgue norm of intermediate derivatives of functions in Sobolev spaces. We present an extension of this estimate into the scale of the general rearrangement-invariant Banach function spaces with the proof based on the Maz'ya's pointwise estimates. As corollaries, we present the Gagliardo--Nirenberg inequality for intermediate derivatives in the case of triples of Orlicz spaces and triples of Lorentz spaces. Finally, we promote the scaling argument to validate the optimality of the Gagliardo-Nirenberg inequality and show that the presented estimate in Orlicz scale is optimal.