Optimal Gagliardo-Nirenberg interpolation inequality for rearrangement invariant spaces

2021/12/21 by Lesnik, Karol, Roskovec, Tomas, Soudsky, Filip
#26D10 #46E15 #46E30 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2112.11570

Abstract

We prove optimality of the Gagliardo-Nirenberg inequality ‖∇ u‖_X\lesssim‖∇² u‖_Y^1/2‖u‖_Z^1/2, where Y, Z are rearrangement invariant Banach function spaces and X=Y^1/2Z^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.

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