Decomposition of L²-Vector Fields on Lipschitz Surfaces: Characterization via Null-Spaces of the Scalar Potential

For ∂Ω the boundary of a bounded and connected strongly Lipschitz domain in Rᵈ with d≥3, we prove that any field f∈ L²(∂Ω ; Rᵈ) decomposes, in a unique way, as the sum of three invisible vector fields---fields whose magnetic potential vanishes in one or both components of Rᵈ∂Ω. Moreover, this decomposition is orthogonal if and only if ∂Ω is a sphere. We also show that any f in L²(∂Ω ; Rᵈ) is uniquely the sum of two invisible fields and a Hardy function, in which case the sum is orthogonal regardless of ∂Ω; we express the corresponding orthogonal projections in terms of layer potentials. When ∂Ω is a sphere, both decompositions coincide and match what has been called the Hardy--Hodge decomposition in the literature.