Fractional differential operators, fractional Sobolev spaces and fractional variation on homogeneous Carnot groups

Abstract

We establish a novel definition of the fractional gradient and divergence of order $$\alpha \in (0, 1)$$ through the use of Riesz potential on homogeneous Carnot groups. We introduce and investigate the distributional fractional Sobolev space and the space of fractional BV functions in this context. Additionally, we provide a definition of fractional Caccioppoli sets on homogeneous Carnot groups and demonstrate their blow-up property, using similar methods as outlined in [13].