Fractional Gagliardo--Nirenberg Inequalities: Pointwise Estimates,Sharp Asymptotics, and Optimal Target Spaces
We establish two pointwise estimates for fractional difference operators, tracking explicitly the dependence of the constants on the smoothness index $s\in(0,1)$. Using these, within the framework of ball Banach function spaces we obtain two fractional Gagliardo--Nirenberg inequalities, including the BMO endpoint case. Furthermore, we establish endpoint asymptotic results as $s\to0^+$ and $s\to1^-$, proving that the asymptotic factors appearing in these inequalities have optimal order. Under the additional assumption that the underlying function space is rearrangement invariant, we show that the optimal Gagliardo--Nirenberg target spaces are precisely those given by the Calder\'on--Lozanovski\u{\i} space. This completely characterizes the rearrangement invariant target spaces for which the corresponding Gagliardo--Nirenberg inequalities hold, thereby answering an open question posed by K. Le\'snik, T. Roskovec, and F. Soudsk\'y. These results can be applied to various function spaces; in particular, they are completely new in the off-diagonal and BMO cases.