# Hardy-Hodge decomposition of vector fields on compact Lipschitz hypersurfaces

Abstract : 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.
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Cited literature [63 references]

https://hal.inria.fr/hal-02936934
Contributor : Laurent Baratchart <>
Submitted on : Friday, September 11, 2020 - 8:10:08 PM
Last modification on : Tuesday, September 22, 2020 - 3:11:02 AM
Long-term archiving on: : Friday, December 4, 2020 - 9:19:53 PM

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• HAL Id : hal-02936934, version 1

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Laurent Baratchart, Dang Pei, Tao Qian. Hardy-Hodge decomposition of vector fields on compact Lipschitz hypersurfaces. 2020. ⟨hal-02936934⟩

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