Inserted: 9 aug 2022
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.