Let F, S be bounded measurable sets in \(\mathbb {R}^d\) . Let \(P_F: L^2(\mathbb {R}^d) \rightarrow L^2(\mathbb {R}^d) \) be the orthogonal projection on the subspace of functions with compact support on F, and let \(B_S: L^2(\mathbb {R}^d) \rightarrow L^2(\mathbb {R}^d)\) be the orthogonal projection on the subspace of functions with Fourier transforms having compact support on S. In this paper, we derive distributional estimates on the eigenvalue sequence \(1 \ge \lambda _1(F,S) \ge \lambda _2(F,S) \ge \cdots > 0\) of the spatio-spectral limiting operator \(B_S P_F B_S: L^2(\mathbb {R}^d) \rightarrow L^2(\mathbb {R}^d)\) . The significance of such estimates lies in their diverse applications in medical imaging, signal processing, geophysics and astronomy. For suitable domains F and S, we prove that \( \# \{ k: \lambda _k(F,S) > \epsilon \} = (2 \pi )^{-d} |F| \cdot |S| + \textrm{Err}(F,S,\epsilon ) \quad \text{ for } \text{ any } \epsilon \in (0,1), \) where \(|F| \cdot |S|\) represents the Lebesgue measure of the domain \(F \times S \subset \mathbb {R}^d \times \mathbb {R}^d\) , and the error term satisfies the following bound: \(\begin{aligned}&|\textrm{Err}(F,S,\epsilon )| \le C_d \frac{\mathcal {H}_{d-1}(\partial F)}{\kappa _{\partial F}} \frac{\mathcal {H}_{d-1}(\partial S)}{\kappa _{\partial S}} \\&\quad \biggl \{ \log \left( \mathcal {H}_{d-1}(\partial F) \mathcal {H}_{d-1}(\partial S)\right) \log (\min \{ \epsilon , 1-\epsilon \}^{-1})^d \\&\quad + \log \left( \mathcal {H}_{d-1}(\partial F) \mathcal {H}_{d-1}(\partial S)\right) ^3 \log (\min \{ \epsilon , 1-\epsilon \}^{-1}) \biggr \} \end{aligned}\) where \(\mathcal {H}_{d-1}(\partial F)\) and \(\mathcal {H}_{d-1}(\partial S)\) denote the \((d-1)\) -dimensional Hausdorff measures of the boundaries of F and S, while \(\kappa _{\partial F}, \kappa _{\partial S} \in (0,1]\) are geometric constants related to an Ahlfors regularity condition on the domain boundaries. When F and S are Euclidean balls, we expect this estimate to be sharp up to logarithmic factors. This improves on recent work of Marceca-Romero-Speckbacher (Arch. Rational Mech. Anal., 2024) which showed that \(|\textrm{Err}(F,S,\epsilon )| \le C_{d,\alpha } \frac{\mathcal {H}_{d-1}(\partial F)}{\kappa _{\partial F}} \frac{\mathcal {H}_{d-1}(\partial S)}{\kappa _{\partial S}} \log \left( \frac{\mathcal {H}_{d-1}(\partial F) \mathcal {H}_{d-1}(\partial S)}{\kappa _{\partial F} \min \{ \epsilon , 1-\epsilon \}}\right) ^{2d(1+\alpha )+1}\) for any \(\alpha >0\) . Our proof is based on the decomposition techniques developed by Marceca-Romero-Speckbacher. The novelty of our approach lies in the use of a two-stage dyadic decomposition with respect to both the spatial and frequency domains, and the application of results in the authors’ prior work on the eigenvalues of spatio-spectral limiting operators associated to cubical domains.
