The Asymptotic Constant for Optimal Integration of C^k Functions on Bounded Lipschitz Domains: Existence and Dependence on the Volume Only

For a bounded open set \(D\subset\mathbb{R}^d\) let \(e_n(C^k(D),\mathrm{INT})\) be the minimal worst-case error of algorithms that use \(n\) function values to integrate, over \(D\), the functions all of whose partial derivatives up to order \(k\) are bounded by \(1\). In Oberwolfach Report 6/2020, Novak asked whether the asymptotic constant \(\limsup_{n}n^{k/d}e_n(C^k(D),\mathrm{INT})\) depends on \(D\), for bounded Lipschitz domains of volume \(1\), and whether it is always a limit. We show that for all \(k,d\ge1\) there is a constant \(c_{k,d}\in(0,\infty)\), defined on the unit cube by the derivatives of order \(k\) alone, such that \(\lim_{n}n^{k/d}e_n(C^k(D),\mathrm{INT})\)\({}=c_{k,d}\,\lambda(D)^{1+k/d}\) for every bounded open set \(D\) with the cone condition, in particular for every bounded Lipschitz domain. So the limit exists, at volume \(1\) it is the same number \(c_{k,d}\) for all such \(D\), and the bounds on the derivatives of order less than \(k\) do not change it. For every bounded open set, without any regularity, the lower limit is at least \(c_{k,d}\lambda(D)^{1+k/d}\). For \(d\ge2\) a hypothesis on the boundary is needed: there are bounded, connected, Jordan measurable “comb” domains of volume \(1\) on which \(n^{k/d}e_n(C^k(D),\mathrm{INT})\)\({}\to\infty\). The proof adapts the scheme of Zador and of Graf and Luschgy from quantization theory: self-similarity on the cube, a cutoff argument that identifies the constants with and without boundary conditions, and a boundary layer that is controlled through the cone condition by local polynomial reproduction. For the Sobolev classes \(L^m_p\) with \(1<p<\infty\) the corresponding asymptotics were proved by Polovinkin (1975, 1989); the case \(p=\infty\) with the full \(C^k\) norm treated here is not covered there. The value of \(c_{k,d}\) and its dependence on \(d\) are not determined. This is an unrefereed note.