Connections between nonlocal operators: from vector calculus identities to a fractional Helmholtz decomposition

Understand

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.