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For $\partial \Omega$ the boundary of a bounded and connected stronglyLipschitz domain in $\mathbb{R}^{d}$ with $d\geq3$, we prove that any field$f\in L^{2} (\partial \Omega ; \mathbb{R}^{d})$ decomposes, in an unique way,as the sum of three silent vector fields---fields whose magnetic potentialvanishes in one or both components of $\mathbb{R}^d\setminus\partial \Omega$.Moreover, this decomposition is orthogonal if and only if $\partial \Omega$ isa sphere. We also show that any $f$ in $L^{2} (\partial \Omega ;\mathbb{R}^{d})$ is uniquely the sum of two silent fields and a Hardy function,in which case the sum is orthogonal regardless of $\partial \Omega$; we expressthe corresponding orthogonal projections in terms of layer potentials. When$\partial \Omega$ is a sphere, both decompositions coincide and match what hasbeen called the Hardy-Hodge decomposition in the literature.

Decomposition of $L^{2}$-vector fields on Lipschitz surfaces: characterization via null-spaces of the scalar potential