Nonlocal Gagliardo-Nirenberg-Sobolev type inequality

We establish Gagliardo-Nirenberg-Sobolev type inequalities on nonlocal Sobolev spaces driven by $p$-L'{e}vy integrable kernels, by imposing some appropriate growth conditions on the associated critical function. This naturally allows to devise Sobolev embeddings, as well as, compact embeddings of 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.