Neural Operators in Anisotropic Fractional Sobolev-Morrey Spaces

This paper develops a comprehensive mathematical theory of anisotropic fractional calculus with mixed regularity structures, addressing fundamental challenges in analyzing high-dimensional functions with heterogeneous smoothness across different coordinate directions. Motivated by applications in scientific machine learning, multiscale analysis, and physical systems with directional preferences, we introduce novel anisotropic fractional Sobolev-Morrey spaces that precisely capture directional scaling behavior through mixed regularity parameters. These spaces provide a refined analytical framework for functions exhibiting varying degrees of smoothness along different coordinates, generalizing classical isotropic theories to anisotropic settings. Our principal contributions establish several sharp functional inequalities: (1) anisotropic Gagliardo-Nirenberg inequalities with mixed fractional derivatives featuring explicit constant dependence on scaling parameters and proven optimality; (2) directional Hardy-Littlewood-Sobolev theory for anisotropic fractional integrals with optimal bounds in Lebesgue and Morrey spaces; (3) compactness criteria in anisotropic function spaces demonstrated through refined real interpolation and harmonic analysis techniques; and (4) optimal approximation rates for deep neural operators in high-dimensional settings, with explicit dimension dependence governed by the anisotropic dimension $d_\alpha = \sum_{i=1}^k \alpha_i^{-1}$. The theoretical framework bridges harmonic analysis, fractional calculus, and deep learning theory, providing rigorous mathematical foundations for understanding the approximation capabilities of modern neural architectures. Furthermore, our results offer principled guidance for neural operator design in scientific computing applications, particularly for problems exhibiting multiscale and anisotropic features. This work opens new research directions in the analysis of partial differential equations, high-dimensional approximation theory, and the mathematical foundations of deep learning.