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In this talk, I would like to present some recent results regarding the behaviour of functions which are uniformly bounded under the action of a certain class of non-convex non-local functionals. In the literature, this class of functionals happens to be a very good substitute of the first order Sobolev spaces. As a consequence various improvements of the classical Poincaré’s inequality, Sobolev’s inequality and Rellich-Kondrachov’s compactness criterion were established. This talk will be focused on addressing the gap between a certain exponential integrability and the boundedness for functions which are finite under the action of these class of non-convex functionals. |