Sobolev spaces on multiple cones

The purpose of this note is to discuss how various Sobolev spaces defined on multiple cones behave with respect to density of smooth functions, interpolation and extension/restriction to/from $\RR^n$. The analysis interestingly combines use of Poincar'e inequalities and of some Hardy type inequalities.

💡 Research Summary

The paper investigates Sobolev spaces defined on a union of several cones sharing a common vertex at the origin, a geometric configuration that naturally arises in many singular or non‑convex domains. Let $C=\bigcup_{i=1}^{m}C_i\subset\mathbb{R}^n$ be the union of $m$ straight cones $C_i$, each with vertex $0$ and opening angle $\alpha_i$, and assume that the cones intersect only at the vertex. For integers $k\ge1$ and $1\le p<\infty$ the authors define the space $W^{k,p}(C)$ as the set of functions whose weak derivatives up to order $k$ belong to $L^{p}(C)$ when restricted to each cone. The central questions are: (1) whether smooth compactly supported functions away from the vertex, $C_c^\infty(C\setminus{0})$, are dense in $W^{k,p}(C)$; (2) whether there exist continuous linear extension and restriction operators between $W^{k,p}(C)$ and the classical Sobolev space $W^{k,p}(\mathbb{R}^n)$; and (3) how interpolation behaves for the scale ${W^{s,p}(C)}_{0\le s\le k}$.

Density of smooth functions.
The authors first establish a Hardy‑type inequality on each cone:
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