We propose two asymptotic expansions of two interrelated integral-type averages, in the context of the fractional ∞-Laplacian Δ∞s for s∈(12,1). This operator has been introduced and first studied in (Comm. Pure Appl. Math. 65 (2012) 337–380). Our expansions are parametrised by the radius of the removed singularity ε, and allow for the identification of Δ∞sϕ(x) as the ε2s-order coefficient of the deviation of the ε-average from the value ϕ(x), in the limit ε→0+. The averages are well posed for functions ϕ that are only Borel regular and bounded.

中文翻译：

---

分数阶无穷拉普拉斯算子的渐近展开

我们针对s∈（12,1）在分数∞-LaplacianΔ∞s的情况下，提出了两个相互关联的积分型平均值的两个渐近展开。已在（Comm。Pure Appl。Math。65（2012）337-380）中引入并对其进行了首次研究。我们的展开是由去除的奇异点ε的半径参数化的，并允许将Δ∞sϕ（x）识别为ε-平均值与ϕ（x）值的偏差的ε2s阶系数。 ε→0 +。平均值是仅由Borel正则和有界的函数构成的。