Fractional Landau Inequalities with Mixed Sobolev Norms and Applications to Multiscale Analysis

This paper develops a comprehensive theory of fractional Landau inequalities with mixed Sobolev norms, extending classical gradient bounds to anisotropic function spaces. Building upon the foundational work of Landau (1925) and recent advances in fractional calculus by Anastassiou (2025), we address the critical limitation of existing theories that operate primarily within isotropic settings. Our framework introduces mixed fractional Sobolev spaces $W^{\nu,p}_\alpha(\mathbb{R}^k)$ that capture directional scaling behavior through parameters $\alpha = (\alpha_1,\dots,\alpha_k)$, enabling precise characterization of functions with heterogeneous regularity across different coordinates. We establish sharp fractional Landau inequalities with constants that explicitly track dependence on both fractional order $\nu$ and anisotropic scaling $\alpha$, proving these bounds through innovative harmonic analysis techniques including directional Littlewood-Paley theory and anisotropic maximal function estimates. The theoretical framework finds compelling applications in neural operator theory, where we prove stability bounds under input perturbations and derive optimal approximation rates for deep networks processing multiscale data. Our results demonstrate that neural operators achieve approximation rates of order $N^{-\nu/d_\alpha}$, where $d_\alpha$ is the anisotropic dimension, substantially improving upon classical isotropic rates when scaling parameters are heterogeneous. This work bridges fractional calculus, harmonic analysis, and deep learning, providing new mathematical foundations for understanding and designing algorithms for high-dimensional, multiscale problems.