Scale-uniform bounded-domain inverse inequality for the full Sobolev range

Establish a general scale-uniform inverse inequality on bounded domains for scaled kernel spaces generated by translates of a reference positive definite kernel, namely prove that, for every bounded domain \(\Omega\), every \(0\leq\alpha\leq\tau\leq\sigma\), and center sets satisfying only \(q_X\leq c_0\delta\), the estimate \(\|v\|_{H^\tau(\Omega)}\leq Cq_X^{\alpha-\tau}\|v\|_{H^\alpha(\Omega)}\) holds for all \(v\in V_{\Phi_\delta,X}\), with \(C\) independent of the kernel scale \(\delta\) as \(\delta\to0\).

Background

The paper proves scale-uniform inverse inequalities for scaled kernel spaces $V_{\Phi_\delta,X}$ under the one-sided scale–separation condition $q_X\leq c_0\delta$. On bounded domains, however, the standard-Sobolev result is obtained only with $L^2(\Omega)$ as the weaker norm, corresponding to $\alpha=0$. In contrast, the whole-space Fourier argument establishes the full range $0\leq\alpha\leq\tau\leq\sigma$.

Extending the whole-space result to bounded domains is difficult because the weaker whole-space Sobolev norm cannot generally be controlled by the corresponding norm of a restriction to $\Omega$. Existing bounded-domain approaches either introduce a negative power of $\delta$, destroying scale uniformity, or, for compactly supported kernels, require the stronger two-sided condition $c_1\delta\leq q_X\leq c_2\delta$, which excludes the multiscale regime $q_X/\delta\to0$.