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Section: New Results

The Hardy-Hodge decomposition

Participant : Laurent Baratchart.

(This is joint work with Qian T. and Dang P. from the university of Macao.) It was proven in previous year that on a smooth compact hypersurface $\Sigma$ embedded in ${ℝ}^n$, a ${ℝ}^n$-valued vector field of $L^p$ class decomposes as the sum of a harmonic gradient from inside $\Sigma$, a harmonic gradient from outside $\Sigma$, and a tangent divergence-free field. This year we extended this result to Lipschitz surfaces for $2-\epsilon <p<2+{\epsilon }^{\text{'}}$, where $\epsilon$ and ${\epsilon }^{\text{'}}$ depend on the Lipschitz constant of the surface. We also proved that the decomposition is valid for $1<p<\infty$ when $\Sigma$ is $VMO$-smooth (i.e. $\Sigma$ is locally the graph of Lipschitz function with derivatives in $VMO$). By projection onto the tangent space, this gives a Hodge decomposition for 1-forms on a Lipschitz surface, which is apparently also new since existing results deal with smooth surfaces (but forms of any degree). This result was reported at the invited session on Harmonic Analysis and Inverse Problems of the Mathematical Congress of the Americas, an article is being written to report on it.