  In this work we further develop a nonlocal calculus theory (initially
introduced in [5]) 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.

Analysis of PDEs