Hardy-Hodge decomposition of vector fields on compact Lipschitz hypersurfaces

For M a compact Lipschitz Riemannian manifold of dimension at least 2, we prove a Helmholtz-Hodge decomposition of tangent $L p$ vector fields as a sum of a gradient and a divergence free fields; the result holds for restricted range of p around 2, and for all $p ∈ (1, ∞)$ when M is V M O-smooth. If, moreover, M is a compact and connected hypersurface having the local Lipschitz graph property, embedded in $R n+1$ with the natural metric, we also establish a Hardy-Hodge decomposition of a $R n+1$-valued vector field of L p class on M as the sum of a tangent divergence free field and of two (traces of) harmonic gradients of Hardy class with exponent p, one from inside and one from outside M. The latter holds for restricted range of p, and for all $p ∈ (1, ∞)$ when M is $C 1$-smooth

On the Recovery of Core and Crustal Components of Geomagnetic Potential Fields

A version with minor modifications is to appear in SIAP (Vol. 77, Issue 5)In Geomagnetism it is of interest to separate the Earth's core magnetic field from the crustal magnetic field. However, measurements by satellites can only sense the sum of the two contributions. In practice, the measured magnetic field is expanded in spherical harmonics and separation into crust and core contribution is achieved empirically, by a sharp cutoff in the spectral domain. In this paper, we derive a mathematical setup in which the two contributions are modeled by harmonic potentials $Φ0$ and $Φ1$ generated on two different spheres $SR 0$ (crust) and $SR 1$ (core) with radii $R1 R0$, we show that it becomes possible if the magnetization m generating $Φ0$ is localized in a strict subregion of $SR 0$. Beyond unique recoverability, we show in this case how to numerically reconstruct characteristic features of $Φ0$ (e.g., spherical harmonic Fourier coefficients). An alternative way of phrasing the results is that knowledge of m on a nonempty open subset of $SR 0$ allows one to perform separation