Nonlocal Green Theorems and Helmholtz Decompositions for Truncated Fractional Gradients

In this work we further develop a nonlocal calculus theory (initially introduced in Bellido et al. (Adv Nonlinear Anal 12:20220316, 2023)) associated with singular fractional-type operators which exhibit kernels with finite support of interactions. The applicability of the framework to nonlocal elasticity and the theory of peridynamics has attracted increased interest and motivation to study it and find connections with its classical counterpart. In particular, a critical contribution of this paper is producing vector identities, integration by part type theorems (such as the Divergence Theorem, Green identities), as well as a Helmholtz–Hodge decomposition. The estimates, together with the analysis performed along the way provide stepping stones for proving additional results in the framework, as well as pathways for numerical implementations.