分数阶Gagliardo–Nirenberg不等式：逐点估计、尖锐渐近行为及最优目标空间
Fractional Gagliardo--Nirenberg Inequalities: Pointwise Estimates,Sharp Asymptotics, and Optimal Target Spaces
AI总结：
该研究针对分数阶差分算子建立逐点估计，推导含BMO端点的分数阶Gagliardo–Nirenberg不等式，确定其渐近因子最优阶，完全刻画重排不变目标空间并解答了开放问题，结果在非对角和BMO情形下为全新成果。
AI中文摘要：
我们针对分数阶差分算子建立了两个逐点估计，明确追踪常数对光滑性指标$s\in(0,1)$的依赖关系。基于这些估计，在球Banach函数空间框架内，我们得到了两个分数阶Gagliardo–Nirenberg不等式，包括BMO端点情形。此外，我们建立了当$s\to0^+$和$s\to1^-$时的端点渐近结果，证明了这些不等式中出现的渐近因子具有最优阶。在基础函数空间为重排不变的额外假设下，我们表明最优Gagliardo–Nirenberg目标空间恰好是Calderón–Lozanovskiĭ空间。这完全刻画了对应Gagliardo–Nirenberg不等式成立的重排不变目标空间，从而回答了K. Leśnik、T. Roskovec和F. Soudský提出的一个开放问题。这些结果可应用于各类函数空间；特别地，它们在非对角和BMO情形下是全新的。
英文摘要：
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 Calderón--Lozanovski\uı 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. Leśnik, T. Roskovec, and F. Soudský. These results can be applied to various function spaces; in particular, they are completely new in the off-diagonal and BMO cases.
