  Nonlocal vector calculus, which is based on the nonlocal forms of gradient,
divergence, and Laplace operators in multiple dimensions, has shown promising
applications in fields such as hydrology, mechanics, and image processing. In
this work, we study the analytical underpinnings of these operators. We
rigorously treat compositions of nonlocal operators, prove nonlocal vector
calculus identities, and connect weighted and unweighted variational
frameworks. We combine these results to obtain a weighted fractional Helmholtz
decomposition which is valid for sufficiently smooth vector fields. Our
approach identifies the function spaces in which the stated identities and
decompositions hold, providing a rigorous foundation to the nonlocal vector
calculus identities that can serve as tools for nonlocal modeling in higher
dimensions.
