Anisotropic Mixed Fractional Landau Inequalities

This paper develops a theory of fractional Landau inequalities in mixed Sobolev norms, extending classical derivative estimates to anisotropic function spaces. We introduce mixed fractional Sobolev spaces \( W_{\alpha}^{\nu,p}(\mathbb{R}^{k}) \), where \( \alpha = (\alpha_1,\dots,\alpha_k) \) encodes directional scaling and characterizes functions with coordinate-dependent regularity. Within this framework, we establish fractional Landau inequalities with constants depending explicitly on the fractional order ν and the anisotropy vector α. The analysis relies on techniques from anisotropic harmonic analysis, including directional Littlewood–Paley decompositions and anisotropic maximal function estimates. The sharpness of the inequalities is discussed, and connections to approximation theory are outlined. These results provide a mathematical bridge between fractional calculus, harmonic analysis, and high-dimensional approximation.