Codimension-independent sharp Sobolev inequality for 1 < p < 2

Improve the codimension-dependent $L^p$-Sobolev estimate for minimal submanifolds in arbitrary codimension when $1<p<2$, with the aim of obtaining the Euclidean sharp Sobolev constant.

Background

The paper discusses earlier work of Balogh, Kristály, and Mester, which established codimension-independent $L^p$-Sobolev inequalities for minimal submanifolds when $2\leq p<n$, but with a constant strictly larger than the Euclidean Aubin–Talenti constant. The authors explain that the corresponding estimate for the range $1$p\geq2$, so the lower-exponent range is not resolved by the results presented in the paper.