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In 2014, Ludwig showed the limiting behavior of the anisotropic Gagliardo$s$-seminorm of a function $f$ as $s\rightarrow 1^-$ and $s\rightarrow0^+$,which extend the results due to Bourgain-Brezis-Mironescu(BBM) andMaz'ya-Shaposhnikova(MS) respectively. Recently, Brezis, Van Schaftingen andYung provided a different approach by replacing the strong $L^p$ norm in theGagliardo $s$-seminorm by the weak $L^p$ quasinorm. They characterized the casefor $s=1$ that complements the BBM formula. The corresponding MS formula for$s=0$ was later established by Yung and the first author. In this paper, wefollow the approach of Brezis-Van Schaftingen-Yung and show the anisotropicversions of $s=1$ and $s=0$. Our result generalizes the work by Brezis, VanSchaftingen, Yung and the first author and complements the work by Ludwig.

Anisotropic versions of the Brezis-Van Schaftingen-Yung approach at $s=1$ and $s=0$