Optimal Gagliardo-Nirenberg interpolation inequality for rearrangement invariant spaces

Karol Lesnik, Tomas Roskovec, Filip Soudsky

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 functio…