This paper reveals the limiting behavior of the fractional Caffarelli–Kohn–Nirenberg inequality in one dimension, highlighting critical issues in identifying extremal functions.
In this paper, we study the fractional Caffarelli–Kohn–Nirenberg (CKN) inequality in one dimension when the parameter γ converges (from the left) to its critical value $1/2$, obtaining Onofri’s inequality in the unit disk as the limit. A difficulty that we encounter is the lack of an explicit expression for the extremal function at which the CKN inequality is attained, which we address by studying solutions of the weighted Liouville equation for the half-Laplacian in dimension one.
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González et al. (2025) studied this question.
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