Brezis-Van Schaftingen-Yung Inequalities Beyond the Classical Setting

In this paper, we extend the framework of Brezis–Van Schaftingen–Yung type inequalities in metric measure spaces by exploring several novel directions. First, we establish finite difference characterizations and fractional Sobolev-type inequalities in settings where the underlying measure is non-doubling or only satisfies a weak doubling condition. Second, we incorporate variable exponent and Orlicz space frameworks to capture nonstandard growth phenomena. Third, we derive anisotropic and directional versions of these inequalities to better address non-isotropic structures, and we apply our results to study regularity properties of nonlocal operators. Finally, we investigate the stability and sharpness of the associated constants as well as interpolation and limiting behaviors that bridge classical and fractional settings. These developments not only generalize existing results but also open new avenues for applications in partial differential equations and numerical analysis.