Going to Lorentz when fractional Sobolev, Gagliardo and Nirenberg estimates fail

TL;DR

This paper develops alternative estimates replacing strong $L^p$ norms with Lorentz norms in fractional Sobolev, Gagliardo-Nirenberg, and Nirenberg inequalities where traditional estimates fail.

Contribution

It introduces new Lorentz norm-based estimates applicable when classical fractional inequalities do not hold.

Findings

Established fractional estimates using Lorentz norms

Extended the applicability of fractional inequalities

Provided alternative bounds in challenging cases

Abstract

In the cases where there is no Sobolev-type or Gagliardo-Nirenberg-type fractional estimate involving $\lvert u\rvert_{W^{s,p}}$, we establish alternative estimates where the strong $L^p$ norms are replaced by Lorentz norms.