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Optimal Gagliardo-Nirenberg interpolation inequality for rearrangement invariant spaces (2112.11570v2)
Published 21 Dec 2021 in math.FA
Abstract: We prove optimality of the Gagliardo-Nirenberg inequality $$ |\nabla u|_{X}\lesssim|\nabla2 u|_Y{1/2}|u|_Z{1/2}, $$ where $Y, Z$ are rearrangement invariant Banach function spaces and $X=Y{1/2}Z{1/2}$ is the Calder\'on--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.