Calculus of Variations and Geometric Measure Theory

G. Foghem

Nonlocal Gagliardo-Nirenberg-Sobolev type inequality

created by foghem on 09 Aug 2022
modified on 19 May 2023


Accepted Paper

Inserted: 9 aug 2022
Last Updated: 19 may 2023

Year: 2021

ArXiv: 2105.07989 PDF


We establish Gagliardo-Nirenberg-Sobolev type inequalities on nonlocal Sobolev spaces driven by $p$-L\'evy integrable functions, by imposing some appropriate growth conditions on the associated critical function. This gives rise to the embedding of the local and the nonlocal Sobolev spaces into Orlicz type spaces. The Gagliardo-Nirenberg-Sobolev type inequalities, as in the classical context, turn out to have some reciprocity with Poincar\'e and Poincar\'e-Sobolev type inequalities. The classical fractional Sobolev inequality is also derived as a direct consequence.