The authors obtain new regularity results in small Lebesgue spaces for the weak or the very weak solution of a linear elliptic equation. The contribution is interesting because the authors improve previous results obtained in the classical Lebesgue spaces. Some of the obtained results have been derived using borderline Sobolev embeddings associated to the Grand Lebesgue spaces. Moreover, another goal is to provide some new Sobolev inclusions using the Generalized Gamma spaces and generalizing the Fusco-Lions-Sbordone results.