Optimal Gagliardo-Nirenberg interpolation inequality for rearrangement invariant spaces

Abstract

We prove optimality of the Gagliardo-Nirenberg inequality $$\Vert \nabla u{\Vert }_X\lesssim \Vert {\nabla }^{2}u{\Vert }_Y^{1/2}\Vert u{\Vert }_Z^{1/2},$$ where $Y,Z$ are rearrangement invariant Banach function spaces and $X=Y^{1/2}{Z}^{1/2}$ is the Calderón--Lozanovskii space. By optimality, we mean that for a certain pair of spaces on the right-hand side, one cannot reduce the space on the left-hand, remaining in the class of rearrangement invariant spaces. The optimality for the Lorentz and Orlicz spaces is given as a consequence, exceeding previous results. We also discuss pointwise inequalities, their importance and counterexample prohibiting an improvement.