Trace estimates with powers of the normal component on rho-convex domains

Determine whether trace estimates of the form \(\int_{\partial R \times S} |v(r,s)|^2 |s\cdot n(r)|^p\,d(r,s)\leq C\|v\|_{hyp}^2\) remain valid for \(\rho\)-convex domains or other special classes of non-smooth domains, for the exponents under consideration in the cited estimate.

Background

The paper proves a trace estimate for spherical kinetic Sobolev spaces on $\rho$-convex domains with the boundary weight $\tau(r,s)|s\cdot n(r)|$. It compares this weight with estimates involving powers of $|s\cdot n(r)|$, which are known under stronger boundary regularity assumptions.

The authors show that the weight in their estimate is generally weaker on $\rho$-convex domains and explicitly leave unresolved whether the stronger trace estimates remain true for $\rho$-convex domains or other selected non-smooth domain classes.