Boundedness and regularity of the Navier–Stokes system in generalized Herz spaces via a novel fractional potential framework

Boundedness and regularity of solutions to the Navier–Stokes system in generalized function space settings remain a challenging task. To address this, tools from harmonic analysis, particularly the boundedness of integral operators, play a crucial role. In this paper, we introduce a new class of generalized fractional potentials that simultaneously incorporate exponential damping and spatial roughness. To the best of our knowledge, this potential has not yet been explored in the existing literature. In the proof of the main result, we employ a combination of refined analytical techniques and impose new appropriate conditions tailored to the generalized setting. The key strategy involves decomposing the summation into several distinct terms, each of which is estimated under specific assumptions. By carefully combining these individual estimates, we establish the boundedness of both the newly defined fractional potential and its classical analogues within the framework of generalized Herz spaces. Furthermore, through a series of remarks, we demonstrate that our generalized potential recovers several well-known operators under particular choices of parameters, thereby showing that our results encompass and extend various existing results in the literature. In addition, by introducing a new technique, we prove that the solution to the Navier–Stokes system remains bounded, which in turn implies regularity. This highlights the broader applicability and strength of the proposed framework in analyzing nonlinear PDEs using harmonic analysis tools.