On asymptotic expansions for the fractional infinity Laplacian

Abstract

We propose two asymptotic expansions of two interrelated integral-type averages, in the context of the fractional $\infty$-Laplacian $\Delta_\infty^s$ for $s\in (\frac{1}{2},1)$. This operator has been introduced and first studied in [Bjorland, C., Caffarelli, L. and Figalli, A., \textsl{Nonlocal Tug-of-War and the inifnity fractional Laplacian}, Comm. Pure Appl. Math., \textbf{65}, pp. 337--380, (2012)]. Our expansions are parametrised by the radius of the removed singularity $\epsilon$, and allow for the identification of $\Delta_\infty^s\phi(x)$ as the $\epsilon^{2s}$-order coefficient of the deviation of the $\epsilon$-average from the value $\phi(x)$, in the limit $\epsilon\to 0+$. The averages are well posed for functions $\phi$ that are only Borel regular and bounded.