Bridging Smoothness and Fractality: A Banach Space Framework for Hybrid Regularity with Spectral and Geometric Applications

We introduce a novel mathematical framework that unifies classical smoothness theory with fractal regularity, addressing a long-standing gap between Sobolev spaces and multifractal structures. This hybrid function space is shown to be complete and to enjoy optimal embedding properties into classical functional settings. A central contribution lies in the local geometric characterization of functions via scale-sensitive dimension metrics, enabling explicit delineation between smooth and fractal behavior. The theory is further strengthened by spectral stability results for operator approximations and by perturbation-resilient estimations of local irregularity. Numerical experiments validate the framework across diverse contexts, including texture segmentation in image analysis, robustness in spectral discretizations, and the modeling of multifractal signals. This work offers a unified analytical and computational toolbox for heterogeneous systems exhibiting abrupt transitions in regularity across space or time.