  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.
