A Remark on the Interpolation Inequality between Sobolev Spaces and Morrey Spaces

Interpolation inequalities play an important role in the study of PDEs and their applications. There are still some interesting open questions and problems related to integral estimates and regularity of solutions to elliptic and/or parabolic equations. The main purpose of our work is to provide an important observation concerning the $L^p$-boundedness property in the context of interpolation inequalities between Sobolev and Morrey spaces, which may be useful for those working in this domain. We also construct a nontrivial counterexample, which shows that the range of admissible values of $p$ is optimal in a certain sense. Our proofs rely on integral representations and on the theory of maximal and sharp maximal functions.