  Interpolation inequalities play an important role in the study of PDEs and
their applications. There are still some interesting open questions and
problems that related to integral estimates and regularity of solutions to the
elliptic and/or parabolic equations. Since many of researchers who are working
on this domain, the main motivation of our work here is to provide an important
observation of $L^p$-boundedness property with respect to the interpolation
inequalities. In this paper, from an idea of integral approximation theory and
Sobolev, Morrey embeddings, we construct a nontrivial counterexample for
interpolation inequalities in the connection of results between Sobolev and
Morrey spaces. Our proofs rely on the integral representation and the theory of
maximal and sharp maximal functions.
